Calculus and analysis

Calculus and analysis

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Delta amplitude

Given a Jacobi amplitude and a elliptic modulus in an elliptic integral,

Abelian integral

An Abelian integral, are also called a hyperelliptic integral, is an integral of the formwhere is a polynomial of degree .

Modified bessel differential equation

The second-order ordinarydifferential equation(1)The solutions are the modified Bessel functions of the first and second kinds, and can be written(2)(3)where is a Bessel function of the first kind, is a Bessel function of the second kind, is a modified Bessel function of the first kind, and is modified Bessel function of the second kind.If , the modified Bessel differential equation becomes(4)which can also be written(5)

Mathieu differential equation

(1)(Abramowitz and Stegun 1972; Zwillinger 1997, p. 125), having solution(2)where and are Mathieu functions. The equation arises in separation of variables of the Helmholtz differential equation in elliptic cylindrical coordinates. Whittaker and Watson (1990) use a slightly different form to define the Mathieu functions.The modified Mathieu differential equation(3)(Iyanaga and Kawada 1980, p. 847; Zwillinger 1997, p. 125) arises in separation of variables of the Helmholtz differential equation in elliptic cylindrical coordinates, and has solutions(4)The associated Mathieu differential equation is given by(5)(Ince 1956, p. 403; Zwillinger 1997, p. 125).

Underdamped simple harmonic motion

Underdamped simple harmonic motion is a special case of dampedsimple harmonic motion(1)in which(2)Since we have(3)it follows that the quantity(4)(5)is positive. Plugging in the trial solution to the differential equation then gives solutions that satisfy(6)i.e., the solutions are of the form(7)Using the Euler formula(8)this can be rewritten(9)We are interested in the real solutions. Since we are dealing here with a linear homogeneous ODE, linear sums of linearly independent solutions are also solutions. Since we have a sum of such solutions in (9), it follows that the imaginary and real parts separately satisfy the ODE and are therefore the solutions we seek. The constant in front of the sine term is arbitrary, so we can identify the solutions as(10)(11)so the general solution is(12)The initial values are(13)(14)so and can be expressed in terms of the initial conditions by(15)(16)The above plot shows an underdamped simple harmonic..

Malmstén's differential equation

The ordinary differential equation(1)It has solution(2)where(3)and is a modified Bessel function of the first kind.

Euler forward method

A method for solving ordinary differential equations using the formulawhich advances a solution from to . Note that the method increments a solution through an interval while using derivative information from only the beginning of the interval. As a result, the step's error is . This method is called simply "the Euler method" by Press et al. (1992), although it is actually the forward version of the analogous Euler backward method.While Press et al. (1992) describe the method as neither very accurate nor very stable when compared to other methods using the same step size, the accuracy is actually not too bad and the stability turns out to be reasonable as long as the so-called Courant-Friedrichs-Lewy condition is fulfilled. This condition states that, given a space discretization, a time step bigger than some computable quantity should not be taken. In situations where this limitation is acceptable, Euler's forward method becomes..

Euler differential equation

The general nonhomogeneous differential equation is given by(1)and the homogeneous equation is(2)(3)Now attempt to convert the equation from(4)to one with constant coefficients(5)by using the standard transformation for linear second-order ordinary differential equations. Comparing (3) and (5), the functions and are(6)(7)Let and define(8)(9)(10)(11)Then is given by(12)(13)(14)which is a constant. Therefore, the equation becomes a second-orderordinary differential equation with constant coefficients(15)Define(16)(17)(18)(19)and(20)(21)The solutions are(22)In terms of the original variable ,(23)Zwillinger (1997, p. 120) gives two other types of equations known as Euler differential equations,(24)(Valiron 1950, p. 201) and(25)(Valiron 1950, p. 212), the latter of which can be solved in terms of Bessel functions...

Third kind

In the theory of special functions, a class of functions is said to be "of the third kind" if it is similar to but distinct from previously defined functions already defined to be of the first and second kinds. The only common functions of the third kind are the elliptic integral of the third kind and the Bessel function of the third kind (more commonly called the Hankel function).

Gardner equation

The partial differential equationwhich can also be rewritten

Inverse scattering method

A method which can be used to solve the initial value problem for certain classes of nonlinear partial differential equations. The method reduces the initial value problem to a linear integral equation in which time appears only implicitly. However, the solutions and various of their derivatives must approach zero as (Infeld and Rowlands 2000).

Benney equation

The Benney equation in dimensions is the nonlinear partial differential equation

Weber differential equations

Consider the differential equation satisfied by(1)where is a Whittaker function, which is given by(2)(3)(Moon and Spencer 1961, p. 153; Zwillinger 1997, p. 128). This is usually rewritten(4)The solutions are parabolic cylinder functions.The equations(5)(6)which arise by separating variables in Laplace's equation in parabolic cylindrical coordinates, are also known as the Weber differential equations. As above, the solutions are known as parabolic cylinder functions.Zwillinger (1997, p. 127) calls(7)the Weber differential equation (Gradshteyn and Ryzhik 2000, p. 989).

Frobenius method

If is an ordinary point of the ordinary differential equation, expand in a Taylor series about . Commonly, the expansion point can be taken as , resulting in the Maclaurin series(1)Plug back into the ODE and group the coefficients by power. Now, obtain a recurrence relation for the th term, and write the series expansion in terms of the s. Expansions for the first few derivatives are(2)(3)(4)(5)(6)If is a regular singular point of the ordinary differential equation,(7)solutions may be found by the Frobenius method or by expansion in a Laurent series. In the Frobenius method, assume a solution of the form(8)so that(9)(10)(11)(12)Now, plug back into the ODE and group the coefficients by power to obtain a recursion formula for the th term, and then write the series expansion in terms of the s. Equating the term to 0 will produce the so-called indicial equation, which will give the allowed values of in the series expansion.As an example, consider the..

Vibration problem

Solution of a system of second-order homogeneous ordinary differential equations with constant coefficients of the formwhere is a positive definite matrix. To solve the vibration problem, 1. Solve the characteristic equation of to get eigenvalues , ..., . Define . 2. Compute the corresponding eigenvectors , ..., . 3. The normal modes of oscillation are given by , ..., , where , ..., and , ..., are arbitrary constants. 4. The general solution is .

Floquet's theorem

Let be a real or complex piecewise-continuous function defined for all values of the real variable and that is periodic with minimum period so that(1)Then the differential equation(2)has two continuously differentiable solutions and , and the characteristic equation is(3)with eigenvalues and . Then Floquet's theorem states that if the roots and are different from each other, then (2) has two linearly independent solutions(4)(5)where and are periodic with period (Magnus and Winkler 1979, p. 4).

Variation of parameters

For a second-order ordinarydifferential equation,(1)Assume that linearly independent solutions and are known to the homogeneous equation(2)and seek and such that(3)(4)Now, impose the additional condition that(5)so that(6)(7)Plug , , and back into the original equation to obtain(8)which simplifies to(9)Combing equations (◇) and (9) and simultaneously solving for and then gives(10)(11)where(12)is the Wronskian, which is a function of only, so these can be integrated directly to obtain(13)(14)which can be plugged in to give the particular solution(15)Generalizing to an th degree ODE, let , ..., be the solutions to the homogeneous ODE and let , ..., be chosen such that(16)and the particular solution is then(17)

Modified spherical bessel differential equation

The modified spherical Bessel differential equation is given by the spherical Bessel differential equation with a negative separation constant,The solutions are called modified spherical Bessel functions of the first and second kinds.

Floquet analysis

Given a system of ordinary differentialequations of the form(1)that are periodic in , the solution can be written as a linear combination of functions of the form(2)where is a function periodic with the same period as the equations themselves. Given an ordinary differential equation of the form(3)where is periodic with period , the ODE has a pair of independent solutions given by the real and imaginary parts of(4)(5)(6)(7)Plugging these into (◇) gives(8)so the real and imaginaryparts are(9)(10)From (◇),(11)(12)(13)Integrating gives(14)where is a constant which must equal 1, so is given by(15)The real solution is then(16)so(17)(18)(19)(20)and(21)(22)(23)(24)which is an integral of motion. Therefore, although is not explicitly known, an integral always exists. Plugging (◇) into (◇) gives(25)which, however, is not any easier to solve than (◇)...

Van der pol equation

The van der Pol equation is an ordinary differential equation that can be derived from the Rayleigh differential equation by differentiating and setting . It is an equation describing self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given byIf , the equation reduces to the equation of simple harmonic motion

Volume element

A volume element is the differential element whose volume integral over some range in a given coordinate system gives the volume of a solid,(1)In , the volume of the infinitesimal -hypercube bounded by , ..., has volume given by the wedge product(2)(Gray 1997).The use of the antisymmetric wedge product instead of the symmetric product is a technical refinement often omitted in informal usage. Dropping the wedges, the volume element for curvilinear coordinates in is given by(3)(4)(5)(6)(7)where the latter is the Jacobian and the are scale factors.

Vardi's integral

Vardi's integral is the beautiful definite integral(1)(2)(3)(OEIS A115252; Gradshteyn and Ryzhik 1980, p. 532; Bailey et al. 2007, p. 160; Moll 2006), where is the gamma function.Other examples similar to these include(4)(5)(Vardi 1988; Gradshteyn and Ryzhik 1980, pp. 571-572).

Upper sum

For a given bounded function over a partition of a given interval, the upper sum is the sum of box areas using the supremum of the function in each subinterval .

Cavalieri's quadrature formula

The definite integral(1)where , , and are real numbers and is the natural logarithm.

Laplace's integral

Laplace's integral is one of the following integral representations of the Legendre polynomial ,(1)(2)It can be evaluated in terms of the hypergeometricfunction.

Singular integral

A singular integral is an integral whose integrand reaches an infinite value at one or more points in the domain of integration. Even so, such integrals can converge, in which case, they are said to exist. (If they do not converge, they are said not to exist.) The most commonly encountered example of a singular integral is the Hilbert transform. (However, note that the logarithmic integral is not singular, since it converges in the classical Riemann sense.)In general, singular integrals can be defined by eliminating a small space including the singularity, and then taking the limit as this small space disappears.

Integration under the integral sign

Integration under the integral sign is the use of the identity(1)to compute an integral. For example, consider(2)for . Multiplying by and integrating between and gives(3)(4)But the left-hand side is equal to(5)so it follows that(6)(Woods 1926, pp. 145-146).

Serret's integral

Serret's integral is given by(1)(2)(OEIS A102886; Serret 1844; Gradshteyn andRyzhik 2000, eqn. 4.291.8; Boros and Moll 2004, p. 243).


An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in physics and elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."The Riemann integral of the function over from to is written(1)Note that if , the integral is written simply(2)as opposed to .Every definition of an integral is based on a particular measure. For instance, the Riemann integral is based on Jordan measure, and the Lebesgue integral is based..

Improper integral

An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. Improper integrals cannot be computed using a normal Riemann integral.For example, the integral(1)is an improper integral. Some such integrals can sometimes be computed by replacing infinite limits with finite values(2)and then taking the limit as ,(3)(4)(5)Improper integrals of the form(6)with one infinite limit and the other nonzero may also be expressed as finite integrals over transformed functions. If decreases at least as fast as , then let(7)(8)(9)(10)and(11)(12)If diverges as for , let(13)(14)(15)(16)(17)and(18)If diverges as for , let(19)(20)(21)and(22)(23)If the integral diverges exponentially, then let(24)(25)(26)and(27)..

Riemann integral

The Riemann integral is the definite integral normally encountered in calculus texts and used by physicists and engineers. Other types of integrals exist (e.g., the Lebesgue integral), but are unlikely to be encountered outside the confines of advanced mathematics texts. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."The Riemann integral is based on the Jordan measure,and defined by taking a limit of a Riemann sum,(1)(2)(3)where and , , and are arbitrary points in the intervals , , and , respectively. The value is called the mesh size of a partition of the interval into subintervals .As an example of the application of the Riemann integral definition, find the area under the curve from 0 to . Divide into segments,..

Repeated integral

A repeated integral is an integral taken multiple times over a single variable (as distinguished from a multiple integral, which consists of a number of integrals taken with respect to different variables). The first fundamental theorem of calculus states that if is the integral of , then(1)Now, if , then(2)It follows by induction that if , then the -fold integral of is given by(3)(4)Similarly, if , then(5)

Area integral

A double integral over three coordinates giving the area within some region ,If a plane curve is given by , then the area between the curve and the x-axis from to is given by

Fubini theorem

Fubini's theorem, sometimes called Tonelli's theorem, establishes a connection between a multiple integral and a repeated one. If is continuous on the rectangular region , then the equalityholds (Thomas and Finney 1996, p. 919).

Path integral

Let be a path given parametrically by . Let denote arc length from the initial point. Then(1)(2)where .

Ahmed's integral

Ahmed's integral is the definite integral(OEIS A096615; Ahmed 2002; Borwein et al.2004, pp. 17-20).This is a special case of a general result that also yields(OEIS A102521 and A098459) as additional cases (Borwein et al. 2004, p. 20), where is Catalan's constant.

Double integral

A double integral is a two-fold multiple integral.Examples of definite double integrals evaluating to simple constants include(1)(2)(3)(4)where is Catalan's constant (Borwein et al. 2004, pp. 48-49), and(5)where is the Euler-Mascheroni constant (Sondow 2003, 2005; Borwein et al. 2004, pp. 48-49).

Abel's integral

Abel's integral is the definite integral(1)(2)(3)(4)(5)(6)(7)(8)(OEIS A102047), where is a cosine integral.

Rogers mod 14 identities

The Rogers mod 14 identities are a set of three Rogers-Ramanujan-likeidentities given by(1)(2)(3)(4)(5)(6)(7)(8)(9)(OEIS A105780, A105781,and A105782).The -identity was found by Rogers (1894) and appears as formula 61 in the list of Slater (1952). The - and -identities were found by Rogers (1917) and appeared as formulas 60 and 59 respectively in Slater (1952).

Ramanujan theta functions

Ramanujan's two-variable theta function is defined by(1)for (Berndt 1985, p. 34; Berndt et al. 2000). It satisfies(2)and(3)(4)(Berndt 1985, pp. 34-35; Berndt et al. 2000), where is a q-Pochhammer symbol, i.e., a q-series.A one-argument form of is also defined by(5)(6)(7)(OEIS A010815; Berndt 1985, pp. 36-37; Berndt et al. 2000), where is a q-Pochhammer symbol. The identities above are equivalent to the pentagonal number theorem.The function also satisfies(8)(9)Ramanujan's -function is defined by(10)(11)(12)(13)(14)(OEIS A000122), where is a Jacobi theta function (Berndt 1985, pp. 36-37). is a generalization of , with the two being connected by(15)Special values of include(16)(17)where is a gamma function.Ramanujan's -function is defined by(18)(19)(20)(21)(22)(23)(OEIS A010054; Berndt 1985, p. 37).Ramanujan's -function is defined by(24)(25)(26)(OEIS A000700; Berndt 1985, p. 37).A..

Fine's equation

The q-series identity(1)(2)(3)(4)where is a q-Pochhammer symbol, is the number of divisors of that are congruent to 1, 5, 7, and 11 (mod 24) minus the number of divisors of congruent to , , , and (mod 24), and is a Kronecker symbol.The coefficients of the first few powers of starting with , 1, ... are 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, ... (OEIS A000377).

Ramanujan psi sum

A sum which includes both the Jacobi triple product and the q-binomial theorem as special cases. Ramanujan's sum iswhere the notation denotes q-series. For , this becomes the q-binomial theorem.

Dyson mod 27 identities

The Dyson mod 27 identities are a set of four Rogers-Ramanujan-likeidentities given by(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(OEIS A104501, A104502,A104503, and A104504).Bailey (1947) systematically studied and generalized Rogers's work on Rogers-Ramanujan type identities in a paper submitted in late 1943. At the time, G. H. Hardy was the editor of the Proceedings of the London Mathematical Society and Hardy had recently taught the young Freeman Dyson in one of his undergraduate classes at Cambridge. He was therefore aware of Dyson's interest in Ramanujan-Rogers-type identities through his rediscovery of the Rogers-Selberg identities. Ignoring the usual convention of keeping the referee anonymous (since as far as Hardy knew, Bailey and Dyson were the only two people in all of England who were interested in Rogers-Ramanujan type identities at the time) and thinking that they would like to be in contact with each..

Dedekind eta function

Min Max The Dedekind eta function is defined over the upper half-plane by(1)(2)(3)(4)(5)(6)(OEIS A010815), where is the square of the nome , is the half-period ratio, and is a q-series (Weber 1902, pp. 85 and 112; Atkin and Morain 1993; Berndt 1994, p. 139).The Dedekind eta function is implemented in the WolframLanguage as DedekindEta[tau].Rewriting the definition in terms of explicitly in terms of the half-period ratio gives the product(7) Min Max Re Im It is illustrated above in the complex plane. is a modular form first introduced by Dedekind in 1877, and is related to the modular discriminant of the Weierstrass elliptic function by(8)(Apostol 1997, p. 47).A compact closed form for the derivative is given by(9)where is the Weierstrass zeta function and and are the invariants corresponding to the half-periods . The derivative of satisfies(10)where is an Eisenstein series, and(11)A special value is given by(12)(13)(OEIS..

Cauchy binomial theorem

(1)(2)where is a q-binomial coefficient.

Quintuple product identity

The quintuple product identity, also called the Watson quintuple product identity, states(1)It can also be written(2)or(3)The quintuple product identity can be written in q-seriesnotation as(4)where and (Gasper and Rahman 1990, p. 134; Leininger and Milne 1999).Using the notation of the Ramanujantheta function (Berndt 1985, p. 83),(5)

Borwein conjectures

Use the definition of the q-series(1)and define(2)Then P. Borwein has conjectured that (1) the polynomials , , and defined by(3)have nonnegative coefficients, (2) the polynomials , , and defined by(4)have nonnegative coefficients, (3) the polynomials , , , , and defined by(5)have nonnegative coefficients, (4) the polynomials , , and defined by(6)have nonnegative coefficients, (5) for odd and , consider the expansion(7)with(8)then if is relatively prime to and , the coefficients of are nonnegative, and (6) given and , consider(9)the generating function for partitions inside an rectangle with hook difference conditions specified by , , and . Let and be positive rational numbers and an integer such that and are integers. then if (with strict inequalities for ) and , then has nonnegative coefficients...

Bailey mod 9 identities

The Bailey mod 9 identities are a set of three Rogers-Ramanujan-like identities appearing as equations (1.6), (1.8), and (1.7) on p. 422 of Bailey (1947) given by(1)(2)(3)(4)(5)(6)(7)(8)(9)(OEIS A104467, A104468,and A104469).Unfortunately, Bailey used non-standard (and essentially unreadable) notation in the paper where these identities first appeared. All three of these identities appear in the list of Slater (1952) as equations (42), (41), and (40) in that order. However, all three contain misprints.In one sense, these identities are the next logical step in the following sequence: 1. The two Rogers-Ramanujan identities (triple product on mod 5 over ). 2. The three Rogers-Selberg identities (triple product on mod 7 over ). 3. The (sort of) four Bailey mod 9 identities (triple product on mod 9 over ). Here, "sort of" refers to the fact that between and , there is an "identity" in which the product side contains..


An antisymmetric tensor of second rank(a.k.a. 2-form).where is the wedge product (or outer product).

Geodesic mapping

A geodesic mapping between two Riemannian manifolds is a diffeomorphism sending geodesics of into geodesics of , whose inverse also sends geodesics to geodesics (Ambartzumian 1982, p. 26).


When the Gaussian curvature is everywhere negative, a surface is called anticlastic and is saddle-shaped. A surface on which is everywhere positive is called synclastic. A point at which the Gaussian curvature is negative is called a hyperbolic point.

Angular acceleration

The angular acceleration is defined as the time derivative of the angular velocity ,

Rodrigues' curvature formula

where is the unit normal vector and is one of the two principal curvatures.

Wiedersehen pair

Two points on a compact Riemann surface such that lies on every geodesic passing through , and conversely. An oriented surface where every point belongs to a Wiedersehen pair is called a Wiedersehen surface.The name is the German word for "seeing again" and was introduced by Blaschke.

Hyperbolic point

A point on a regular surface is said to be hyperbolic if the Gaussian curvature or equivalently, the principal curvatures and , have opposite signs.

Principal radius of curvature

At each point on a given a two-dimensional surface, there are two "principal" radii of curvature. The larger is denoted , and the smaller . The "principal directions" corresponding to the principal radii of curvature are perpendicular to one another. In other words, the surface normal planes at the point and in the principal directions are perpendicular to one another, and both are perpendicular to the surface tangent plane at the point.

Third fundamental form

Let be a regular surface with points in the tangent space of . Then the third fundamental form is given bywhere is the shape operator.

Geodesic curvature

For a unit speed curve on a surface, the length of the surface-tangential component of acceleration is the geodesic curvature . Curves with are called geodesics. For a curve parameterized as , the geodesic curvature is given bywhere , , and are coefficients of the first fundamental form and are Christoffel symbols of the second kind.

Tangent developable

A ruled surface is a tangent developable of a curve if can be parameterized by . A tangent developable is a flat surface.

Principal curvatures

The maximum and minimum of the normal curvature and at a given point on a surface are called the principal curvatures. The principal curvatures measure the maximum and minimum bending of a regular surface at each point. The Gaussian curvature and mean curvature are related to and by(1)(2)This can be written as a quadratic equation(3)which has solutions(4)(5)

Gauss's theorema egregium

Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-space may be understood intrinsically to that surface. "Residents" of the surface may observe the Gaussian curvature of the surface without ever venturing into full three-dimensional space; they can observe the curvature of the surface they live in without even knowing about the three-dimensional space in which they are embedded.In particular, Gaussian curvature can be measured by checking how closely the arc lengths of circles of small radii correspond to what they should be in Euclidean space, . If the arc length of circles tends to be smaller than what is expected in Euclidean space, then the space is positively curved; if larger, negatively; if the same, 0 Gaussian curvature.Gauss (effectively) expressed the theorema egregium by saying that the Gaussian curvature at a point is given by where is the Riemann tensor, and and are an orthonormal..


A surface on which the Gaussian curvature is everywhere positive. When is everywhere negative, a surface is called anticlastic. A point at which the Gaussian curvature is positive is called an elliptic point.

Planar point

A point on a regular surface is said to be planar if the Gaussian curvature and (where is the shape operator), or equivalently, both of the principal curvatures and are 0.


A patch (also called a local surface) is a differentiable mapping , where is an open subset of . More generally, if is any subset of , then a map is a patch provided that can be extended to a differentiable map from into , where is an open set containing . Here, (or more generally, ) is called the map trace of .

Gauss equations

If is a regular patch on a regular surface in with normal , then(1)(2)(3)where , , and are coefficients of the second fundamental form and are Christoffel symbols of the second kind.

Parabolic point

A point on a regular surface is said to be parabolic if the Gaussian curvature but (where is the shape operator), or equivalently, exactly one of the principal curvatures equals 0.

Surface area

Surface area is the area of a given surface. Roughly speaking, it is the "amount" of a surface (i.e., it is proportional to the amount of paint needed to cover it), and has units of distance squared. Surface area is commonly denoted for a surface in three dimensions, or for a region of the plane (in which case it is simply called "the" area).The following tables gives lateral surface areas for some common surfaces. Here, denotes the radius, the height, the ellipticity of a spheroid, the base perimeter, the slant height, the tube radius of a torus, and the radius from the rotation axis of the torus to the center of the tube (Beyer 1987). Note that many of these surfaces are surfaces of revolution, for which Pappus's centroid theorem can often be used to easily compute the surface area.surfaceconeconical frustumcubecylinderoblate spheroidprolate spheroidpyramidpyramidal frustumspherespherical lunetoruszoneEven simple..

Fundamental forms

There are three types of so-called fundamental forms. The most important are the first and second (since the third can be expressed in terms of these). The fundamental forms are extremely important and useful in determining the metric properties of a surface, such as line element, area element, normal curvature, Gaussian curvature, and mean curvature. Let be a regular surface with points in the tangent space of . Then the first fundamental form is the inner product of tangent vectors,(1)For , the second fundamental form is the symmetric bilinear form on the tangent space ,(2)where is the shape operator. The third fundamental form is given by(3)The first and secondfundamental forms satisfy(4)(5)where is a regular patch and and are the partial derivatives of with respect to parameters and , respectively. Their ratio is simply the normal curvature(6)for any nonzero tangent vector. The third fundamentalform is given in terms of the first and..

Support function

Let be an oriented regular surface in with normal . Then the support function of is the function defined by

Normal developable

A ruled surface is a normal developable of a curve if can be parameterized by , where is the normal vector (Gray 1993, pp. 352-354; first edition only).

First fundamental form

Let be a regular surface with points in the tangent space of . Then the first fundamental form is the inner product of tangent vectors,(1)The first fundamental form satisfies(2)The first fundamental form (or line element) is givenexplicitly by the Riemannian metric(3)It determines the arc length of a curve on a surface.The coefficients are given by(4)(5)(6)The coefficients are also denoted , , and . In curvilinear coordinates (where ), the quantities(7)(8)are called scale factors.

Striction curve

A noncylindrical ruled surface alwayshas a parameterization of the form(1)where , , and is called the striction curve of . Furthermore, the striction curve does not depend on the choice of the base curve. The striction and director curves of the helicoid(2)are(3)(4)For the hyperbolic paraboloid(5)the striction and director curves are(6)(7)

Normal curvature

Let be a unit tangent vector of a regular surface . Then the normal curvature of in the direction is(1)where is the shape operator. Let be a regular surface, , be an injective regular patch of with , and(2)where . Then the normal curvature in the direction is(3)where , , and are the coefficients of the first fundamental form and , , and are the coefficients of the second fundamental form.The maximum and minimum values of the normal curvature at a point on a regular surface are called the principal curvatures and .

Shape operator

The negative derivative(1)of the unit normal vector field of a surface is called the shape operator (or Weingarten map or second fundamental tensor). The shape operator is an extrinsic curvature, and the Gaussian curvature is given by the determinant of . If is a regular patch, then(2)(3)At each point on a regular surface , the shape operator is a linear map(4)The shape operator for a surface is given by the Weingartenequations.

Lyapunov function

A Lyapunov function is a scalar function defined on a region that is continuous, positive definite, for all ), and has continuous first-order partial derivatives at every point of . The derivative of with respect to the system , written as is defined as the dot product(1)The existence of a Lyapunov function for which on some region containing the origin, guarantees the stability of the zero solution of , while the existence of a Lyapunov function for which is negative definite on some region containing the origin guarantees the asymptotical stability of the zero solution of .For example, given the system(2)(3)and the Lyapunov function , we obtain(4)which is nonincreasing on every region containing the origin, and thus the zero solution is stable.

Erfc differential equation

The second-order ordinarydifferential equation(1)whose solutions may be written either(2)where is the repeated integral of the erfc function (Abramowitz and Stegun 1972, p. 299), or(3)where is a Hermite polynomial and is a confluent hypergeometric function of the first kind.

Ellipsoidal wave equation

The ordinary differential equationwhere is a Jacobi elliptic function (Arscott 1981).

Struve differential equation

The ordinary differential equationwhere is the gamma function (Abramowitz and Stegun 1972, p. 496; Zwillinger 1997, p. 127). The solution iswhere and are Bessel functions of the first and second kinds, and is a Struve function (Abramowitz and Stegun 1972).

Lommel differential equation

The Lommel differential equation is a generalization of the Besseldifferential equation given by(1)or, in the most general form, by(2)The case is the most common (Watson 1966, p. 345; Zwillinger 1997, p. 125; Gradshteyn and Ryzhik 2000, p. 937), and its solutions are given by(3)(4)where are Lommel functions. Note that is most commonly written simply as .The second-order ordinarydifferential equation(5)is sometimes also called the Lommel differential equation.

Liouville's equation

The second-order ordinarydifferential equation(1)is called Liouville's equation (Goldstein and Braun 1973; Zwillinger 1997, p. 124),as are the partial differential equations(2)(Matsumo 1987; Zwillinger 1997, p. 133) and(3)(Calogero and Degasperis 1982, p. 60; Zwillinger 1997, p. 133).

Duffing differential equation

The most general forced form of the Duffing equation is(1)Depending on the parameters chosen, the equation can take a number of special forms. For example, with no damping and no forcing, and taking the plus sign, the equation becomes(2)(Bender and Orszag 1978, p. 547; Zwillinger 1997, p. 122). This equation can display chaotic behavior. For , the equation represents a "hard spring," and for , it represents a "soft spring." If , the phase portrait curves are closed.If instead we take , , reset the clock so that , and use the minus sign, the equation is then(3)This can be written as a system of first-order ordinary differential equations as(4)(5)(Wiggins 1990, p. 5) which, in the unforced case, reduces to(6)(7)(Wiggins 1990, p. 6; Ott 1993, p. 3).The fixed points of this set of coupled differential equations are given by(8)so , and(9)(10)giving . The fixed points are therefore , , and .Analysis..

Differential algebra

Differential algebra is a field of mathematics that attempts to use methods from abstract algebra to study solutions of systems of polynomial nonlinear ordinary and partial differential equations. It is a generalization of classical commutative algebra and is primarily based on the work of Ritt (1950) and Kolchin (1973). Mansfield (1991) gave a terminating algorithm for differential Gröbner bases, which are differential analogs of polynomial Gröbner bases.

Spherical bessel differential equation

Take the Helmholtz differential equation(1)in spherical coordinates. This is just Laplace's equation in spherical coordinates with an additional term,(2)Multiply through by ,(3)This equation is separable in . Call the separation constant ,(4)Now multiply through by ,(5)This is the spherical Bessel differential equation. It can be transformed by letting , then(6)Similarly,(7)so the equation becomes(8)Now look for a solution of the form , denoting a derivative with respect to by a prime,(9)(10)(11)so(12)(13)(14)(15)(16)But the solutions to this equation are Bessel functionsof half integral order, so the normalized solutions to the original equation are(17)which are known as spherical Bessel functions. The two types of solutions are denoted (spherical Bessel function of the first kind) or (spherical Bessel function of the second kind), and the general solution is written(18)where(19)(20)..

Legendre differential equation

The Legendre differential equation is the second-orderordinary differential equation(1)which can be rewritten(2)The above form is a special case of the so-called "associated Legendre differential equation" corresponding to the case . The Legendre differential equation has regular singular points at , 1, and .If the variable is replaced by , then the Legendre differential equation becomes(3)derived below for the associated () case.Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind. If is an integer, the function of the first kind reduces to a polynomial known as the Legendre polynomial.The Legendre differential equation can be solved using the Frobenius method..

Simple harmonic motion--quadratic perturbation

Given a simple harmonic oscillator with a quadratic perturbation, write the perturbation term in the form ,(1)find the first-order solution using a perturbation method. Write(2)and plug back into (1) and group powers to obtain(3)To solve this equation, keep terms only to order and note that, because this equation must hold for all powers of , we can separate it into the two simultaneous differential equations(4)(5)Setting our clock so that , the solution to (4) is then(6)Plugging this solution back into (5) then gives(7)The equation can be solved to give(8)Combining and then gives(9)(10)where the sinusoidal and cosinusoidal terms of order (from the ) have been ignored in comparison with the larger terms from .As can be seen in the top figure above, this solution approximates only for . As the lower figure shows, the differences from the unperturbed oscillator grow stronger over time for even relatively small values of ...

Simple harmonic motion

Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is executed by any quantity obeying the differential equation(1)where denotes the second derivative of with respect to , and is the angular frequency of oscillation. This ordinary differential equation has an irregular singularity at . The general solution is(2)(3)where the two constants and (or and ) are determined from the initial conditions.Many physical systems undergoing small displacements, including any objects obeying Hooke's law, exhibit simple harmonic motion. This equation arises, for example, in the analysis of the flow of current in an electronic CL circuit (which contains a capacitor and an inductor). If a damping force such as Friction is present, an additional term must be added to the differential equation and motion dies out over time...

Sharpe's differential equation

A generalization of the Bessel differentialequation for functions of order 0, given bySolutions arewhere is a confluent hypergeometric function of the first kind.

Damped simple harmonic motion

Adding a damping force proportional to to the equation of simple harmonic motion, the first derivative of with respect to time, the equation of motion for damped simple harmonic motion is(1)where is the damping constant. This equation arises, for example, in the analysis of the flow of current in an electronic CLR circuit, (which contains a capacitor, an inductor, and a resistor). The curve produced by two damped harmonic oscillators at right angles to each other is called a harmonograph, and simplifies to a Lissajous curve if .The damped harmonic oscillator can be solved by looking for trial solutions of the form . Plugging this into (1) gives(2)(3)This is a quadratic equation with solutions(4)There are therefore three solution regimes depending on the signof the quantity inside the square root,(5)The three regimes are summarized in the following table.regimeunderdampingcritical dampingoverdampingIf a periodic (sinusoidal) forcing..

Separation of variables

Separation of variables is a method of solving ordinary and partial differential equations.For an ordinary differential equation(1)where is nonzero in a neighborhood of the initial value, the solution is given implicitly by(2)If the integrals can be done in closed form and the resulting equation can be solved for (which are two pretty big "if"s), then a complete solution to the problem has been obtained. The most important equation for which this technique applies is , the equation for exponential growth and decay (Stewart 2001).For a partial differential equation in a function and variables , , ..., separation of variables can be applied by making a substitution of the form(3)breaking the resulting equation into a set of independent ordinary differential equations, solving these for , , ..., and then plugging them back into the original equation.This technique works because if the product of functions of independent variables..

Lamé's differential equation types

Whittaker and Watson (1990, pp. 539-540) write Lamé's differential equation for ellipsoidal harmonics of the first kind of the four types as(1)(2)(3)(4)where(5)(6)

Lamé's differential equation

The ordinary differential equation(1)(Byerly 1959, p. 255). The solution is denoted and is known as an ellipsoidal harmonic of the first kind, or Lamé function. Whittaker and Watson (1990, pp. 554-555) give the alternative forms (2)(3)(4)(5)(6)(Whittaker and Watson 1990, pp. 554-555; Ward 1987; Zwillinger 1997, p. 124). Here, is a Weierstrass elliptic function, is a Jacobi elliptic function, and(7)(8)(9)Two other equations named after Lamé are given by(10)and(11)(Moon and Spencer 1961, p. 157; Zwillinger 1997, p. 124).

Laguerre differential equation

The Laguerre differential equation is given by(1)Equation (1) is a special case of the more general associatedLaguerre differential equation, defined by(2)where and are real numbers (Iyanaga and Kawada 1980, p. 1481; Zwillinger 1997, p. 124) with .The general solution to the associated equation (2) is(3)where is a confluent hypergeometric function of the first kind and is a generalized Laguerre polynomial.Note that in the special case , the associated Laguerre differential equation is of the form(4)so the solution can be found using an integratingfactor(5)(6)(7)(8)as(9)(10)(11)where is the En-function.The associated Laguerre differential equation has a regular singular point at 0 and an irregular singularity at . It can be solved using a series expansion, (12)(13)(14)(15)(16)This requires(17)(18)for . Therefore,(19)for , 2, ..., so(20)(21)(22)If is a nonnegative integer, then the series terminates and the..

D'alembert's solution

The method of d'Alembert provides a solution to the one-dimensional waveequation(1)that models vibrations of a string.The general solution can be obtained by introducing new variables and , and applying the chain rule to obtain(2)(3)(4)(5)Using (4) and (5) to compute the left andright sides of (3) then gives(6)(7)(8)(9)respectively, so plugging in and expanding then gives(10)This partial differential equation has general solution(11)(12)where and are arbitrary functions, with representing a right-traveling wave and a left-traveling wave.The initial value problem for a string located at position as a function of distance along the string and vertical speed can be found as follows. From the initial condition and (12),(13)Taking the derivative with respect to then gives(14)(15)and integrating gives(16)Solving (13) and (16) simultaneously for and immediately gives(17)(18)so plugging these into (13) then gives the solution..

Lagrange's equation

The partial differential equation(Gray 1997, p. 399), whose solutions are called minimal surfaces. This corresponds to the mean curvature equalling 0 over the surface.d'Alembert's equationis sometimes also known as Lagrange's equation (Zwillinger 1997, pp. 120 and 265-268).

D'alembert's equation

The ordinary differential equationwhere and and are given functions. This equation is sometimes also known as Lagrange's equation (Zwillinger 1997).

Critically damped simple harmonic motion

Critical damping is a special case of dampedsimple harmonic motion(1)in which(2)where is the damping constant. Therefore(3)In this case, so the solutions of the form satisfy(4)One of the solutions is therefore(5)In order to find the other linearly independent solution, we can make use of the identity(6)Since we have , simplifies to . Equation (6) therefore becomes(7)The general solution is therefore(8)In terms of the constants and , the initial values are(9)(10)so(11)(12)The above plot shows a critically damped simple harmonic oscillator with , for a variety of initial conditions .For sinusoidally forced simple harmonic motion with critical damping, the equation of motion is(13)and the Wronskian is(14)(15)Plugging this into the equation for the particular solution gives(16)(17)Applying the harmonic addition theoremthen gives(18)where(19)..

Kovalevskaya top equations

The system of ordinary differentialequations(1)(2)

Confluent hypergeometric differential equation

The second-order ordinarydifferential equationsometimes also called Kummer's differential equation (Slater 1960, p. 2; Zwillinger 1997, p. 124). It has a regular singular point at 0 and an irregular singularity at . The solutionsare called confluent hypergeometric function of the first and second kinds, respectively. Note that the confluent hypergeometric function of the first kind is also denoted or .

Indirectly conformal mapping

An indirectly conformal mapping, sometimes called an anticonformal mapping, is a mapping that reverses all angles, whereas an isogonal mapping can reverse some angles and preserve others.For example, if is a conformal map, then is an indirectly conformal map, and is an isogonal mapping.


The symbol used by engineers and some physicists to denote i, the imaginary number . is probably preferred over because the symbol (or ) is commonly used to denote current.

Real axis

The real axis is the line in the complex plane corresponding to zero imaginary part, . Every real number corresponds to a unique point on the real axis.

Multivariable calculus

Multivariable calculus is the branch of calculus that studies functions of more than one variable. Partial derivatives and multiple integrals are the generalizations of derivative and integral that are used. An important theorem in multivariable calculus is Green's theorem, which is a generalization of the first fundamental theorem of calculus to two dimensions.


A counterexample is a form of counter proof.Given a hypothesis stating that is true for all , show that there exists a such that is false, contradicting the hypothesis.

Monotonic sequence

A sequence such that either (1) for every , or (2) for every .

Logarithmic derivative

The logarithmic derivative of a function is defined as the derivative of the logarithm of a function. For example, the digamma function is defined as the logarithmic derivative of the gamma function,


The smallest value of a set, function, etc. The minimum value of a set of elements is denoted or , and is equal to the first element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the minimum is 1. The maximum and minimum are the simplest order statistics.The minimum value of a variable is commonly denoted (cf. Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 84). In this work, the convention is used.The minimum of a set of elements is implemented in the Wolfram Language as Min[list] and satisfies the identities(1)(2)A continuous function may assume a minimum at a single point or may have minima at a number of points. A global minimum of a function is the smallest value in the entire range of the function, while a local minimum is the smallest value in some local neighborhood.For a function which is continuous at a point , a necessary but not sufficient condition for to have a local..

Weierstrass substitution

The Weierstrass substitution is the trigonometric substitution which transforms an integral of the forminto one of the formAccording to Spivak (2006, pp. 382-383), this is undoubtably the world's sneakiest substitution.The Weierstrass substitution can also be useful in computing a Gröbner basis to eliminate trigonometric functions from a system of equations (Trott 2006, p. 39).

Box integral

A box integral for dimension with parameters and is defined as the expectation of distance from a fixed point of a point chosen at random over the unit -cube,(1)(Bailey et al. 2006).Two special cases include(2)(3)which, with , correspond to hypercube point picking (to a fixed vertex) and hypercube line picking, respectively.Hypercube point picking to the center isgiven by(4)

Unit square integral

Integrals over the unit square arising in geometricprobability are(1)which give the average distances in square point picking from a point picked at random in a unit square to a corner and to the center, respectively.Unit square integrals involving the absolute valueare given by(2)(3)for and , respectively.Another simple integral is given by(4)(Bailey et al. 2007, p. 67). Squaring the denominator gives(5)(6)(7)(8)(9)(OEIS A093754; M. Trott, pers. comm.), where is Catalan's constant and is a generalized hypergeometric function. A related integral is given by(10)which diverges in the Riemannian sense, as can quickly seen by transforming to polar coordinates. However, taking instead Hadamard integral to discard the divergent portion inside the unit circle gives(11)(12)(13)(14)(OEIS A093753), where is Catalan's constant.A collection of beautiful integrals over the unit squareare given by Guillera and Sondow..

Dual bivector

A dual bivector is defined byand a self-dual bivector by

Mean curvature

Let and be the principal curvatures, then their mean(1)is called the mean curvature. Let and be the radii corresponding to the principal curvatures, then the multiplicative inverse of the mean curvature is given by the multiplicative inverse of the harmonic mean,(2)In terms of the Gaussian curvature ,(3)The mean curvature of a regular surface in at a point is formally defined as(4)where is the shape operator and denotes the matrix trace. For a Monge patch with ,(5)(Gray 1997, p. 399).If is a regular patch, then the mean curvature is given by(6)where , , and are coefficients of the first fundamental form and , , and are coefficients of the second fundamental form (Gray 1997, p. 377). It can also be written(7)Gray (1997, p. 380).The Gaussian and mean curvature satisfy(8)with equality only at umbilic points, since(9)If is a point on a regular surface and and are tangent vectors to at , then the mean curvature of at is related to the..

Euler curvature formula

The curvature of a surface satisfieswhere is the normal curvature in a direction making an angle with the first principal direction and and are the principal curvatures.

Map trace

Let a patch be given by the map , where is an open subset of , or more generally by , where is any subset of . Then (or more generally, ) is called the trace of .

Elliptic point

A point on a regular surface is said to be elliptic if the Gaussian curvature or equivalently, the principal curvatures and have the same sign.

Second fundamental form

Let be a regular surface with points in the tangent space of . For , the second fundamental form is the symmetric bilinear form on the tangent space ,(1)where is the shape operator. The second fundamental form satisfies(2)for any nonzero tangent vector.The second fundamental form is given explicitly by(3)where(4)(5)(6)and are the direction cosines of the surface normal. The second fundamental form can also be written(7)(8)(9)(10)(11)(12)(13)(14)where is the normal vector, is a regular patch, and and are the partial derivatives of with respect to parameters and , respectively, or(15)(16)(17)

Line of curvature

A curve on a surface whose tangents are always in the direction of principalcurvature. The equation of the lines of curvature can be writtenwhere and are the coefficients of the first and second fundamental forms.

Radius of curvature

The radius of curvature is given by(1)where is the curvature. At a given point on a curve, is the radius of the osculating circle. The symbol is sometimes used instead of to denote the radius of curvature (e.g., Lawrence 1972, p. 4).Let and be given parametrically by(2)(3)then(4)where and . Similarly, if the curve is written in the form , then the radius of curvature is given by(5)In polar coordinates , the radius of curvature is given by(6)where and (Gray 1997, p. 89).


(1)where is the radius vector and is the derivative with respect to time. Expressed in terms of the arc length,(2)where is the unit tangent vector, so the speed (which is the magnitude of the velocity) is(3)

Principal curve

A curve on a regular surface is a principal curve iff the velocity always points in a principal direction, i.e.,where is the shape operator and is a principal curvature. If a surface of revolution generated by a plane curve is a regular surface, then the meridians and parallels are principal curves.

Total curvature

The term "total curvature" is used in two different ways in differential geometry.The total curvature, also called the third curvature, of a space curve with line elements , , and along the normal, tangent, and binormal vectors respectively, is defined as the quantity(1)(2)where is the curvature and is the torsion (Kreyszig 1991, p. 47). The term is apparently also applied to the derivative directly , namely(3)(Kreyszig 1991, p. 47).The second use of "total curvature" is as a synonym for Gaussiancurvature (Kreyszig 1991, p. 131).

Poincaré's theorem

If (i.e., is an irrotational field) in a simply connected neighborhood of a point , then in this neighborhood, is the gradient of a scalar field ,for , where is the gradient operator. Consequently, the gradient theorem givesfor any path located completely within , starting at and ending at .This means that if , the line integral of is path-independent.


The torsion of a space curve, sometimes also called the "second curvature" (Kreyszig 1991, p. 47), is the rate of change of the curve's osculating plane. The torsion is positive for a right-handed curve, and negative for a left-handed curve. A curve with curvature is planar iff .The torsion can be defined by(1)where is the unit normal vector and is the unit binormal vector. Written explicitly in terms of a parameterized vector function ,(2)(3)(Gray 1997, p. 192), where denotes a scalar triple product and is the radius of curvature.The quantity is called the radius of torsion and is denoted or .

Harmonic analysis

In music, if a note has frequency , integer multiples of that frequency, and so on, are known as harmonics. As a result, the mathematical study of overlapping waves is called harmonic analysis.Harmonic analysis is a diverse field including such branches as Fourier series, isospectral manifolds (hearing the shape of a drum), and topological groups. Signal processing, medical imaging, and quantum mechanics are three of the fields that use harmonic analysis extensively.

Haar measure

Any locally compact Hausdorff topological group has a unique (up to scalars) nonzero left invariant measure which is finite on compact sets. If the group is Abelian or compact, then this measure is also right invariant and is known as the Haar measure.More formally, let be a locally compact group. Then a left invariant Haar measure on is a Borel measure satisfying the following conditions: 1. for every and every measurable . 2. for every nonempty open set . 3. for every compact set . For example, the Lebesgue measure is an invariantHaar measure on real numbers.In addition, if is an (algebraic) group, then with the discrete topology is a locally compact group. A left invariant Haar measure on is the counting measure on .

Subharmonic function

Let be an open set and a real-valued continuous function on . Suppose that for each closed disk and every real-valued harmonic function defined on a neighborhood of which satisfies on , it holds that on the open disk . Then is said to be subharmonic on (Krantz 1999, p. 99).1. If are subharmonic on , then so is . 2. If is subharmonic on and is a constant, than is subharmonic on . 3. If are subharmonic on , then is also subharmonic on .

Kelvin transformation

Let be a domain in for . Then the transformationonto a domain , whereis called a Kelvin transformation. If is a harmonic function on , then is also harmonic on .

Harnack's principle

Let be harmonic functions on a connected open set . Then either uniformly on compact sets or there is a finite-values harmonic function on such that uniformly on compact sets.

Dirichlet problem

The problem of finding the connection between a continuous function on the boundary of a region with a harmonic function taking on the value on . In general, the problem asks if such a solution exists and, if so, if it is unique. The Dirichlet problem is extremely important in mathematical physics (Courant and Hilbert 1989, pp. 179-180 and 240; Logan 1997; Krantz 1999b).If is a continuous function on the boundary of the open unit disk , then define(1)where is the boundary of . Then is continuous on the closed unit disk and harmonic on (Krantz 1999a, p. 93).For the case of rational boundary data without poles, the resulting solution of the Dirichlet problem is also rational (Ebenfelt and Viscardi 2005), the proof of which led to Viscardi winning the 2005-2006 Siemens-Westinghouse competition (Siemens Foundation 2005; Mathematical Association of America 2006)...

Harnack's inequality

Let be an open disk, and let be a harmonic function on such that for all . Then for all , we have

Dirichlet energy

Let be a real-valued harmonic function on a bounded domain , then the Dirichlet energy is defined as , where is the gradient.

Harmonic function

Any real function with continuous second partial derivatives which satisfies Laplace's equation,(1)is called a harmonic function. Harmonic functions are called potential functions in physics and engineering. Potential functions are extremely useful, for example, in electromagnetism, where they reduce the study of a 3-component vector field to a 1-component scalar function. A scalar harmonic function is called a scalar potential, and a vector harmonic function is called a vector potential.To find a class of such functions in the plane, write the Laplace's equation in polar coordinates(2)and consider only radial solutions(3)This is integrable by quadrature, so define ,(4)(5)(6)(7)(8)(9)so the solution is(10)Ignoring the trivial additive and multiplicative constants, the general pure radial solution then becomes(11)(12)Other solutions may be obtained by differentiation, such as(13)(14)(15)(16)and(17)Harmonic functions..

Green's function--poisson's equation

Poisson's equation is(1)where is often called a potential function and a density function, so the differential operator in this case is . As usual, we are looking for a Green's function such that(2)But from Laplacian,(3)so(4)and the solution is(5)Expanding in the spherical harmonics gives(6)where and are greater than/less than symbols. this expression simplifies to(7)where are Legendre polynomials, and . Equations (6) and (7) give the addition theorem for Legendre polynomials.In cylindrical coordinates, the Green'sfunction is much more complicated,(8)where and are modified Bessel functions of the first and second kinds (Arfken 1985).

Green's function--helmholtz differential equation

The inhomogeneous Helmholtz differentialequation is(1)where the Helmholtz operator is defined as . The Green's function is then defined by(2)Define the basis functions as the solutions to the homogeneous Helmholtz differential equation(3)The Green's function can then be expanded in terms of the s,(4)and the delta function as(5)Plugging (◇) and (◇) into (◇) gives(6)Using (◇) gives(7)(8)This equation must hold true for each , so(9)(10)and (◇) can be written(11)The general solution to (◇) is therefore(12)(13)

Green's function

Generally speaking, a Green's function is an integral kernel that can be used to solve differential equations from a large number of families including simpler examples such as ordinary differential equations with initial or boundary value conditions, as well as more difficult examples such as inhomogeneous partial differential equations (PDE) with boundary conditions. Important for a number of reasons, Green's functions allow for visual interpretations of the actions associated to a source of force or to a charge concentrated at a point (Qin 2014), thus making them particularly useful in areas of applied mathematics. In particular, Green's function methods are widely used in, e.g., physics, and engineering.More precisely, given a linear differential operator acting on the collection of distributions over a subset of some Euclidean space , a Green's function at the point corresponding to is any solution of(1)where denotes the delta..

Differential equation

A differential equation is an equation that involves the derivatives of a function as well as the function itself. If partial derivatives are involved, the equation is called a partial differential equation; if only ordinary derivatives are present, the equation is called an ordinary differential equation. Differential equations play an extremely important and useful role in applied math, engineering, and physics, and much mathematical and numerical machinery has been developed for the solution of differential equations.

Universal differential equation

A universal differential equation (UDE) is a nontrivial differential-algebraic equation with the property that its solutions approximate to arbitrary accuracy any continuous function on any interval of the real line.Rubel (1981) found the first known UDE by showing that, given any continuous function and any positive continuous function , there exists a solution of(1)such that(2)for all .Duffin (1981) found two additional families of UDEs,(3)and(4)whose solutions are for .Briggs (2002) found a further family of UDEs given by(5)for .

Delay differential equation

A delay differential equation (also called a differential delay equation or difference-differential equation, although the latter term has a different meaning in the modern literature) is a special type of functional differential equation. Delay differential equations are similar to ordinary differential equations, but their evolution involves past values of the state variable. The solution of delay differential equations therefore requires knowledge of not only the current state, but also of the state a certain time previously.Examples include the equations defining the Dickmanfunction(1)and the Buchstab function(2)(Panario 1998).

Buchstab function

The Buchstab function is defined by the delay differential equation(1)(Panario 1998). It approaches the asymptotic value as (and in fact has nearly reached this value already by ).

Residue theorem

An analytic function whose Laurent series is given by(1)can be integrated term by term using a closed contour encircling ,(2)(3)The Cauchy integral theorem requires thatthe first and last terms vanish, so we have(4)where is the complex residue. Using the contour gives(5)so we have(6)If the contour encloses multiple poles, then the theorem gives the general result(7)where is the set of poles contained inside the contour. This amazing theorem therefore says that the value of a contour integral for any contour in the complex plane depends only on the properties of a few very special points inside the contour.The diagram above shows an example of the residue theorem applied to the illustrated contour and the function(8)Only the poles at 1 and are contained in the contour, which have residues of 0 and 2, respectively. The values of the contour integral is therefore given by(9)..

Principal part

If a function has a pole at , then the negative power part(1)of the Laurent series of about (2)is called the principal part of at . For example, the principal part of(3)is (Krantz 1999, pp. 46-47).

Critical index

Let be the Maclaurin series of a meromorphic function with a finite or infinite number of poles at points , indexed so thatthen a pole will occur as many times in the sequence as indicated by its order. Any index such thatholds is then called a critical index of (Henrici 1988, pp. 641-642).

Complex residue

The constant in the Laurent series(1)of about a point is called the residue of . If is analytic at , its residue is zero, but the converse is not always true (for example, has residue of 0 at but is not analytic at ). The residue of a function at a point may be denoted . The residue is implemented in the Wolfram Language as Residue[f, z, z0].Two basic examples of residues are given by and for .The residue of a function around a point is also defined by(2)where is counterclockwise simple closed contour, small enough to avoid any other poles of . In fact, any counterclockwise path with contour winding number 1 which does not contain any other poles gives the same result by the Cauchy integral formula. The above diagram shows a suitable contour for which to define the residue of function, where the poles are indicated as black dots.It is more natural to consider the residue of a meromorphic one-form because it is independent of the choice of coordinate. On a Riemann..

Del bar operator

The operator is defined on a complex manifold, and is called the 'del bar operator.' The exterior derivative takes a function and yields a one-form. It decomposes as(1)as complex one-forms decompose into complexform of type(2)where denotes the direct sum. More concretely, in coordinates ,(3)and(4)These operators extend naturally to forms of higher degree. In general, if is a -complex form, then is a -form and is a -form. The equation expresses the condition of being a holomorphic function. More generally, a -complex form is called holomorphic if , in which case its coefficients, as written in a coordinate chart, are holomorphic functions.The del bar operator is also well-defined on bundle sections of a holomorphic vector bundle. The reason is because a change in coordinates or trivializations is holomorphic...

Zalcman's lemma

Let be a family of meromorphic functions on the unit disk which are not normal at 0. Then there exist sequences in , , , and a nonconstant function meromorphic in the plane such thatlocally and uniformly (in the spherical sense) in the complex plane (Schwick 2000), where and .

Montel's theorem

Let be an analytic function of , regular in the half-strip defined by and . If is bounded in and tends to a limit as for a certain fixed value of between and , then tends to this limit on every line in , and uniformly for .

Complex rotation

A complex rotation is a map of the form , where is a real number, which corresponds to counterclockwise rotation by radians about the origin of points the complex plane.

Lusin area integral

If is a domain and is a one-to-one analytic function, then is a domain, and(Krantz 1999, p. 150).

Logarithmic branch point

A branch point whose neighborhood of values wrap around an infinite number of times as their complex arguments are varied. The point under the function is therefore a logarithmic branch point. Logarithmic branch points are equivalent to logarithmic singularities.

Complex magnification

A complex magnification is a map of the form , where is a positive real number, which corresponds to magnification about the origin of points in the complex plane by the factor if is greater than 1, or shrinking by a factor if is less than 1.

Complex form

The differential forms on decompose into forms of type , sometimes called -forms. For example, on , the exterior algebra decomposes into four types:(1)(2)where , , and denotes the direct sum. In general, a -form is the sum of terms with s and s. A -form decomposes into a sum of -forms, where .For example, the 2-forms on decompose as(3)(4)The decomposition into forms of type is preserved by holomorphic functions. More precisely, when is holomorphic and is a -form on , then the pullback is a -form on .Recall that the exterior algebra is generated by the one-forms, by wedge product and addition. Then the forms of type are generated by(5)The subspace of the complex one-forms can be identified as the -eigenspace of the almost complex structure , which satisfies . Similarly, the -eigenspace is the subspace . In fact, the decomposition of determines the almost complex structure on .More abstractly, the forms into type are a group representation of , where..

Complex conjugate

The complex conjugate of a complex number is defined to be(1)The conjugate matrix of a matrix is the matrix obtained by replacing each element with its complex conjugate, (Arfken 1985, p. 210).The complex conjugate is implemented in the WolframLanguage as Conjugate[z].Note that there are several notations in common use for the complex conjugate. Applied physics and engineering texts tend to prefer , while most modern math and theoretical physics texts favor . Unfortunately, the notation is also commonly used to denote adjoint operators matrices. Because of these mutually contradictory conventions, care is needed when consulting the literature. In this work, is used to denote the complex conjugate.Common notational conventions for complex conjugate are summarized in the table below.notationreferencesThis work; Abramowitz and Stegun (1972, p. 16), Anton (2000, p. 528), Harris and Stocker (1998, p. 21),..

Complex analysis

Complex analysis is the study of complex numbers together with their derivatives, manipulation, and other properties. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems. Contour integration, for example, provides a method of computing difficult integrals by investigating the singularities of the function in regions of the complex plane near and between the limits of integration.The key result in complex analysis is the Cauchy integral theorem, which is the reason that single-variable complex analysis has so many nice results. A single example of the unexpected power of complex analysis is Picard's great theorem, which states that an analytic function assumes every complex number, with possibly one exception, infinitely often in any neighborhood of an essential singularity!A fundamental result of complex analysis is the Cauchy-Riemann..

Symmetric points

Two points and are symmetric with respect to a circle or straight line if all circles and straight lines passing through and are orthogonal to . Möbius transformations preserve symmetry. Let a straight line be given by a point and a unit vector , thenwhere is the complex conjugate. Let a circle be given by center and radius , then

Quadratic surd

A number of the form , where is a positive rational number which is not the square of another rational number is called a pure quadratic surd. A number of the form , where is rational and is a pure quadratic surd is sometimes called a mixed quadratic surd (Hardy 1967, p. 20).Quadratic surds are sometimes also called quadratic irrationals.In 1770, Lagrange proved that any quadratic surd has a regular continued fraction which is periodic after some point. This result is known as Lagrange's continued fraction theorem.

Laplace transform

The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.The (unilateral) Laplace transform (not to be confused with the Lie derivative, also commonly denoted ) is defined by(1)where is defined for (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by "the" Laplace transform, although a bilateral Laplace transform is sometimes also defined as(2)(Oppenheim et al. 1997). The unilateral Laplace transform is implemented in the Wolfram Language as LaplaceTransform[f[t], t, s] and the inverse Laplace transform as InverseRadonTransform.The inverse Laplace transform is known as the Bromwich integral, sometimes known as the Fourier-Mellin integral..


A power is an exponent to which a given quantity is raised. The expression is therefore known as " to the th power." A number of powers of are plotted above (cf. Derbyshire 2004, pp. 68 and 73).The power may be an integer, real number, or complex number. However, the power of a real number to a non-integer power is not necessarily itself a real number. For example, is real only for .A number other than 0 taken to the power 0 is defined to be 1, which followsfrom the limit(1)This fact is illustrated by the convergence of curves at in the plot above, which shows for , 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number gives smaller and smaller numbers that approach one from above, while doing the same with a number between 0 and 1 gives larger and larger numbers that approach one from below. For square roots, the total power taken is , which approaches 0 as is large, giving in the limit that..


The hyperfactorial (Sloane and Plouffe 1995) is the function defined by(1)(2)where is the K-function.The hyperfactorial is implemented in the WolframLanguage as Hyperfactorial[n].For integer values , 2, ... are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (OEIS A002109).The hyperfactorial can also be generalized to complex numbers, as illustrated above.The Barnes G-function and hyperfactorial satisfy the relation(3)for all complex .The hyperfactorial is given by the integral(4)and the closed-form expression(5)for , where is the Riemann zeta function, its derivative, is the Hurwitz zeta function, and(6) also has a Stirling-like series(7)(OEIS A143475 and A143476). has the special value(8)(9)(10)where is the Euler-Mascheroni constant and is the Glaisher-Kinkelin constant.The derivative is given by(11)..

Bessel polynomial

Krall and Fink (1949) defined the Bessel polynomials as the function(1)(2)where is a modified Bessel function of the second kind. They are very similar to the modified spherical bessel function of the second kind . The first few are(3)(4)(5)(6)(7)(OEIS A001497). These functions satisfy thedifferential equation(8)Carlitz (1957) subsequently considered the related polynomials(9)This polynomial forms an associated Sheffer sequencewith(10)This gives the generating function(11)The explicit formula is(12)(13)where is a double factorial and is a confluent hypergeometric function of the first kind. The first few polynomials are(14)(15)(16)(17)(OEIS A104548).The polynomials satisfy the recurrence formula(18)

Faà di bruno's formula

Faà di Bruno's formula gives an explicit equation for the th derivative of the composition . If and are functions for which all necessary derivatives are defined, then(1)where and the sum is over all partitions of , i.e., values of , ..., such that(2)(Roman 1980).It can also be expressed in terms of Bell polynomial as(3)(M. Alekseyev, pers. comm., Nov. 3, 2006).Faà di Bruno's formula can be cast in a framework that is a special case of a Hopf algebra (Figueroa and Gracia-Bondía 2005).The first few derivatives for symbolic and are given by(4)(5)(6)

Lebesgue measure

The Lebesgue measure is an extension of the classical notions of length and area to more complicated sets. Given an open set containing disjoint intervals, the Lebesgue measure is defined byGiven a closed set ,A unit line segment has Lebesgue measure 1; the Cantor set has Lebesgue measure 0. The Minkowski measure of a bounded, closed set is the same as its Lebesgue measure (Ko 1995).

Lebesgue integral

The Lebesgue integral is defined in terms of upper and lower bounds using the Lebesgue measure of a set. It uses a Lebesgue sum where is the value of the function in subinterval , and is the Lebesgue measure of the set of points for which values are approximately . This type of integral covers a wider class of functions than does the Riemann integral.The Lebesgue integral of a function over a measure space is writtenor sometimesto emphasize that the integral is taken with respect to the measure .

Lebesgue integrable

A nonnegative measurable function is called Lebesgue integrable if its Lebesgue integral is finite. An arbitrary measurable function is integrable if and are each Lebesgue integrable, where and denote the positive and negative parts of , respectively.The following equivalent characterization of Lebesgue integrable follows as a consequence of monotone convergence theorem. A nonnegative measurable function is Lebesgue integrable iff there exists a sequence of nonnegative simple functions such that the following two conditions are satisfied: 1. . 2. almost everywhere.

Lebesgue's dominated convergence theorem

Suppose that is a sequence of measurable functions, that pointwise almost everywhere as , and that for all , where is integrable. Then is integrable, and

Lebesgue decomposition

Any complex measure decomposes into an absolutely continuous measure and a singular measure , with respect to some positive measure . This is the Lebesgue decomposition,

Darboux integral

The Darboux integral, also called a Darboux-Stieltjes integral, is a variant of the Stieltjes integral that is defined as a common value for the lower and upper Darboux integrals.Let and be bounded real functions on an interval , with nondecreasing. For any partition given by , let .The lower Darboux integral is the supremum of all lower sums of the formwhere denotes the infimum of over the interval .Likewise the upper Darboux integral is the infimum ofall upper sums of the formwhere denotes the supremum of over the interval .The lower Darboux integral is less or equal to the upper Darboux integral, and that the Darboux integral is a linear form on the vector space of Darboux-integrable functions on for a given .If , the original upper and lower Darboux integrals proposed by Darboux in 1875 are recovered.If the Stieltjes integral exists, then the Darboux integral also exists and has the same value. If is continuous, then the two integrals are identical...

Product measure

Let and be measure spaces, let be the collection of all measurable rectangles contained in , and let be the premeasure defined on byfor . By the product measure , one means the Carathéodory extension of defined on the sigma-algebra of -measurable subsets of where denotes the outer measure induced by the premeasure on .


Let be a collection of subsets of a set and let be a set function. The function is called a premeasure provided that is finitely additive, countably monotone, and that if , where is the empty set.

Polar representation

A polar representation of a complex measure is analogous to the polar representation of a complex number as , where ,(1)The analog of absolute value is the total variation , and is replaced by a measurable real-valued function . Or sometimes one writes with instead of .More precisely, for any measurable set ,(2)where the integral is the Lebesgue integral. It is natural to extend the definition of the Lebesgue integral to complex measures using the polar representation(3)

Jordan measure decomposition

If is a real measure (i.e., a measure that takes on real values), then one can decompose it according to where it is positive and negative. The positive variation is defined by(1)where is the total variation. Similarly, the negative variation is(2)Then the Jordan decomposition of is defined as(3)When already is a positive measure then . More generally, if is absolutely continuous, i.e.,(4)then so are and . The positive and negative variations can also be written as(5)and(6)where is the decomposition of into its positive and negative parts.The Jordan decomposition has a so-called minimum property. In particular, given any positive measure , the measure has another decomposition(7)The Jordan decomposition is minimal with respect to these changes. One way to say this is that any decomposition must have and ...

Correlation dimension

Define the correlation integral as(1)where is the Heaviside step function. When the below limit exists, the correlation dimension is then defined as(2)If is the correlation exponent, then(3)It satisfies(4)where is the capacity dimension and is the information dimension (correcting the typo in Baker and Gollub 1996), and is conjectured to be equal to the Lyapunov dimension.To estimate the correlation dimension of an -dimensional system with accuracy requires data points, where(5)where is the length of the "plateau region." If an attractor exists, then an estimate of saturates above some given by(6)which is sometimes known as the fractal Whitney embedding prevalence theorem.

Pointwise convergence

The hypothesis is that, for is a measure space, for each , as . The hypothesis may be weakened to almost everywhere convergence.

Jordan measure

Let be a bounded set in the plane, i.e., is contained entirely within a rectangle. The outer Jordan measure of is the greatest lower bound of the areas of the coverings of , consisting of finite unions of rectangles. The inner Jordan measure of is the difference between the area of an enclosing rectangle , and the outer measure of the complement of in . The Jordan measure, when it exists, is the common value of the outer and inner Jordan measures of .If is a bounded nonnegative function on the interval , the ordinate set of f is the setThen is Riemann integrable on iff is Jordan measurable, in which case the Jordan measure of is equal to .There are analogous versions of Jordan measure in all other dimensions.

Complex measure

A measure which takes values in the complex numbers. The set of complex measures on a measure space forms a vector space. Note that this is not the case for the more common positive measures. Also, the space of finite measures () has a norm given by the total variation measure , which makes it a Banach space.Using the polar representation of , it is possible to define the Lebesgue integral using a complex measure,Sometimes, the term "complex measure" is used to indicate an arbitrary measure. The definitions for measure can be extended to measures which take values in any vector space. For instance in spectral theory, measures on , which take values in the bounded linear maps from a Hilbert space to itself, represent the operator spectrum of an operator.

Pincherle derivative

Let , then for any operator ,is called the Pincherle derivative of . If is a shift-invariant operator, then its Pincherle derivative is also a shift-invariant operator.

Measure theory

Measure theory is the study of measures. It generalizes the intuitive notions of length, area, and volume. The earliest and most important examples are Jordan measure and Lebesgue measure, but other examples are Borel measure, probability measure, complex measure, and Haar measure.

Hamburger moment problem

A necessary and sufficient condition that there should exist at least one nondecreasing function such thatfor , 1, 2, ..., with all the integrals converging, is that sequence is positive definite (Widder 1941, p. 129).

Univalent function

A function or transformation in which does not overlap .In modular function theory, a function is called univalent on a subgroup if it is automorphic under and valence 1 (Apostol 1997).

Fundamental theorem of symmetric functions

Any symmetric polynomial (respectively, symmetric rational function) can be expressed as a polynomial (respectively, rational function) in the elementary symmetric polynomials on those variables.There is a generalization of this theorem to polynomial invariants of permutation groups , which states that any polynomial invariant can be represented as a finite linear combination of special -invariant orbit polynomials with symmetric functions as coefficients, i.e.,where ,and , ..., are elementary symmetric functions, and , ..., are special terms. Furthermore, any special term has a total degree , and a maximal variable degree .

Orthonormal functions

A pair of functions and are orthonormal if they are orthogonal and each normalized so that(1)(2)These two conditions can be succinctly written as(3)where is a weighting function and is the Kronecker delta.

Orthogonal functions

Two functions and are orthogonal over the interval with weighting function if(1)If, in addition,(2)(3)the functions and are said to be orthonormal.

Function space

is the collection of all real-valued continuous functions defined on some interval . is the collection of all functions with continuous th derivatives. A function space is a topological vector space whose "points" are functions.


Let be a set. An operation on is a function from a power of into . More precisely, given an ordinal number , a function from into is an -ary operation on . If is a finite ordinal, then the -ary operation is a finitary operation on .

Odd function

A univariate function is said to be odd provided that . Geometrically, such functions are symmetric about the origin. Examples of odd functions include , , the sine , hyperbolic sine , tangent , hyperbolic tangent , error function erf , inverse erf , and the Fresnel integrals , and .An even function times an odd function is odd, and the product of two odd functions is even while the sum or difference of two nonzero functions is odd if and only if each summand function is odd. The product and quotient of two odd functions is an even function.If an even function is differentiable, then its derivative is an odd function; what's more, if an odd function is integrable, then its integral over a symmetric interval , , is identically zero. Similarly, if an even function is differentiable, then its derivative is an odd function while the integral of such a function over a symmetric interval is twice the value of its integral over the interval .Ostensibly, one can define..

Function centroid

By analogy with the geometric centroid, the centroid of an arbitrary function is defined as(1)where the integrals are taken over the domain of . For example, for the Gaussian function , the centroid is(2)If is normalized so that(3)then its centroid is equivalent to its mean.

Symmetric function

A symmetric function on variables , ..., is a function that is unchanged by any permutation of its variables. In most contexts, the term "symmetric function" refers to a polynomial on variables with this feature (more properly called a "symmetric polynomial"). Another type of symmetric functions is symmetric rational functions, which are the rational functions that are unchanged by permutation of variables.The symmetric polynomials (respectively, symmetric rational functions) can be expressed as polynomials (respectively, rational functions) in the elementary symmetric polynomials. This is called the fundamental theorem of symmetric functions.A function is sometimes said to be symmetric about the y-axis if . Examples of such functions include (the absolute value) and (the parabola)...


A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from to is an object such that every is uniquely associated with an object . A function is therefore a many-to-one (or sometimes one-to-one) relation. The set of values at which a function is defined is called its domain, while the set of values that the function can produce is called its range. Here, the set is called the codomain of .In the context of univariate, real-valued functions , the fact that domain elements are mapped to unique range elements can be expressed graphically by way of the vertical line test.In some literature, the term "map" is synonymous with function. Some caution must be exhibited, however, as it is not uncommon for the term map to denote a function with some sort of unspoken regularity assumption, e.g., in point-set topology, where "map" sometimes refers to a function which is continuous..


Let be a function defined on a set and taking values in a set . Then is said to be a surjection (or surjective map) if, for any , there exists an for which . A surjection is sometimes referred to as being "onto."Let the function be an operator which maps points in the domain to every point in the range and let be a vector space with . Then a transformation defined on is a surjection if there is an such that for all .In the categories of sets, groups, modules, etc., an epimorphism is the same as a surjection, and is used synonymously with "surjection" outside of category theory.

Strictly increasing function

A function is said to be strictly increasing on an interval if for all , where . On the other hand, if for all , the function is said to be (nonstrictly) increasing.

Strictly decreasing function

A function is said to be strictly decreasing on an interval if for all , where . On the other hand, if for all , the function is said to be (nonstrictly) decreasing.

Negative part

Let , then the negative part of is the function defined byNote that the negative part is itself a nonnegative function. The negative part satisfies the identitywhere is the positive part of .

Smooth function

A smooth function is a function that has continuous derivatives up to some desired order over some domain. A function can therefore be said to be smooth over a restricted interval such as or . The number of continuous derivatives necessary for a function to be considered smooth depends on the problem at hand, and may vary from two to infinity. A function for which all orders of derivatives are continuous is called a C-infty-function.

Natural boundary

Consider a power series in a complex variable (1)that is convergent within the open disk . Convergence is limited to within by the presence of at least one singularity on the boundary of . If the singularities on are so densely packed that analytic continuation cannot be carried out on a path that crosses , then is said to form a natural boundary (or "natural boundary of analyticity") for the function .As an example, consider the function(2)Then formally satisfies the functional equation(3)The series (◇) clearly converges within . Now consider . Equation (◇) tells us that which can only be satisfied if . Considering now , equation (◇) becomes and hence . Substituting for in equation (◇) then gives(4)from which it follows that(5)Now consider equal to any of the fourth roots of unity, , , for example . Then . Applying this procedure recursively shows that is infinite for any such that with , 1, 2, .... In any arc of..

Even function

A univariate function is said to be even provided that . Geometrically, such functions are symmetric about the -axis. Examples of even functions include 1 (or, in general, any constant function), , , , and .An even function times an odd function is odd, while the sum or difference of two nonzero functions is even if and only if each summand function is even. The product or quotient of two even functions is again even.If a univariate even function is differentiable, then its derivative is an odd function; what's more, if an even function is integrable, then its integral over a symmetric interval , , is precisely the same as twice the integral over the interval . Similarly, if an odd function is differentiable, then its derivative is an even function while the integral of such a function over a symmetric interval is identically zero.Ostensibly, one can define a similar notion for multivariate functions by saying that such a function is even if and only ifEven..

Euler's homogeneous function theorem

Let be a homogeneous function of order so that(1)Then define and . Then(2)(3)(4)Let , then(5)This can be generalized to an arbitrary number of variables(6)where Einstein summation has been used.

Multivalued function

A multivalued function, also known as a multiple-valued function (Knopp 1996, part 1 p. 103), is a "function" that assumes two or more distinct values in its range for at least one point in its domain. While these "functions" are not functions in the normal sense of being one-to-one or many-to-one, the usage is so common that there is no way to dislodge it. When considering multivalued functions, it is therefore necessary to refer to usual "functions" as single-valued functions.While the trigonometric, hyperbolic, exponential, and integer power functions are all single-valued functions, their inverses are multivalued. For example, the function maps each complex number to a well-defined number , while its inverse function maps, for example, the value to . While a unique principal value can be chosen for such functions (in this case, the principal square root is the positive one), the choices cannot..

Elementary function

A function built up of a finite combination of constant functions, field operations (addition, multiplication, division, and root extractions--the elementary operations)--and algebraic, exponential, and logarithmic functions and their inverses under repeated compositions (Shanks 1993, p. 145; Chow 1999). Among the simplest elementary functions are the logarithm, exponential function (including the hyperbolic functions), power function, and trigonometric functions.Following Liouville (1837, 1838, 1839), Watson (1966, p. 111) defines the elementarytranscendental functions as(1)(2)(3)and lets , etc.Not all functions are elementary. For example, the normaldistribution function(4)(5)is a notorious example of a nonelementary function, where is erf (sometimes known as the error function). The elliptic integral(6)is another, where is an elliptic integral of the first kind...

Nyquist frequency

In order to recover all Fourier components of a periodic waveform, it is necessary to use a sampling rate at least twice the highest waveform frequency. The Nyquist frequency, also called the Nyquist limit, is the highest frequency that can be coded at a given sampling rate in order to be able to fully reconstruct the signal, i.e.,

Directional derivative

The directional derivative is the rate at which the function changes at a point in the direction . It is a vector form of the usual derivative, and can be defined as(1)(2)where is called "nabla" or "del" and denotes a unit vector.The directional derivative is also often written in the notation(3)(4)where denotes a unit vector in any given direction and denotes a partial derivative.Let be a unit vector in Cartesian coordinates, so(5)then(6)

Vector derivative

A vector derivative is a derivative taken with respect to a vector field. Vector derivatives are extremely important in physics, where they arise throughout fluid mechanics, electricity and magnetism, elasticity, and many other areas of theoretical and applied physics.The following table summarizes the names and notations for various vector derivatives.symbolvector derivativegradientLaplacian or vector Laplacian or directional derivativedivergencecurlconvective derivativeVector derivatives can be combined in different ways, producing sets of identities that are also very important in physics.Vector derivative identities involving the curl include(1)(2)(3)(4)(5)In Cartesian coordinates(6)(7)In spherical coordinates,(8)(9)(10)Vector derivative identities involving the divergenceinclude(11)(12)(13)(14)(15)In Cartesian coordinates,(16)(17)(18)(19)(20)(21)In spherical coordinates,(22)(23)(24)(25)(26)(27)By..

Surface integral

For a scalar function over a surface parameterized by and , the surface integral is given by(1)(2)where and are tangent vectors and is the cross product.For a vector function over a surface, the surfaceintegral is given by(3)(4)(5)where is a dot product and is a unit normal vector. If , then is given explicitly by(6)If the surface is surface parameterized using and , then(7)

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