The Reuleaux tetrahedron, sometimes also called the spherical tetrahedron, is the three-dimensional solid common to four spheres of equal radius placed so that the center of each sphere lies on the surface of the other three. The centers of the spheres are therefore located at the vertices of a regular tetrahedron, and the solid consists of an "inflated" tetrahedron with four curved edges.Note that the name, coined here for the first time, is based on the fact that the geometric shape is the three-dimensional analog of the Reuleaux triangle, not the fact that it has constant width. In fact, the Reuleaux tetrahedron is not a solid of constant width. However, Meißner (1911) showed how to modify the Reuleaux tetrahedron to form a surface of constant width by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. Depending on which three edge arcs are replaced (three that have a common..
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 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).
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 .
The definite integral(1)where , , and are real numbers and is the natural logarithm.
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.
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 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 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..
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)..
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,..
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)
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'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).
Let be a path given parametrically by . Let denote arc length from the initial point. Then(1)(2)where .
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.
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 is the definite integral(1)(2)(3)(4)(5)(6)(7)(8)(OEIS A102047), where is a cosine integral.
Picking two independent sets of points and from a unit uniform distribution and placing them at coordinates gives points uniformly distributed over the unit square.The distribution of distances from a randomly selected point in the unit square to its center is illustrated above.The expected distance to the square's center is(1)(2)(3)(4)(Finch 2003, p. 479; OEIS A103712), where is the universal parabolic constant. The expected distance to a fixed vertex is given by(5)(6)which is exactly twice .The expected distances from the closest and farthest vertices are given by(7)(8)Pick points at randomly in a unit square and take the convex hull . Let be the expected area of , the expected perimeter, and the expected number of vertices of . Then(9)(10)(11)(12)(13)(14)(OEIS A096428 and A096429), where is the multiplicative inverse of Gauss's constant, is the gamma function, and is the Euler-Mascheroni constant (Rényi and Sulanke..
Let two points and be picked randomly from a unit -dimensional hypercube. The expected distance between the points , i.e., the mean line segment length, is then(1)This multiple integral has been evaluated analytically only for small values of . The case corresponds to the line line picking between two random points in the interval .The first few values for are given in the following table.OEIS1--0.3333333333...2A0915050.5214054331...3A0730120.6617071822...4A1039830.7776656535...5A1039840.8785309152...6A1039850.9689420830...7A1039861.0515838734...8A1039871.1281653402...The function satisfies(2)(Anderssen et al. 1976), plotted above together with the actual values.M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the -dimensional integral to an integral over a 1-dimensional integrand such that(3)The first few values are(4)(5)(6)(7)In the limit as , these..
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..
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).
Let and be nonnegative and continuous functions on the closed interval , then the solid of revolution obtained by rotating the curves and about the -axis from to and taking the region enclosed between them has volume given by
Let be a plane region bounded above by a continuous curve , below by the -axis, and on the left and right by and , then the volume of the solid of revolution obtained by rotating about the -axis is given by
Let be a nonnegative and continuous function on the closed interval , then the solid of revolution obtained by rotating the curve about the -axis from to has volume given by
Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle. This problem is not affine, so a simple formula in terms of the area or linear properties of the original triangle apparently does not exist.However, if the original triangle is chosen to be an isosceles right triangle with unit legs, then the average length of a line with endpoints chosen at random inside it is given by(1)(2)(3)(OEIS A093063; M. Trott, pers. comm., Mar. 10, 2004), which is numerically surprisingly close to .Similarly, if the original triangle is chosen to be an equilateral triangle with unit side lengths, then the average length of a line with endpoints chosen at random inside it is given by(4)(5)The integrand can be split up into the four pieces(6)(7)(8)(9)As illustrated above, symmetry immediately gives and , so(10)With some effort, the integrals and can be done analytically to give..
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)
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..
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 following vector integrals are related to the curltheorem. If(1)then(2)If(3)then(4)The following are related to the divergence theorem.If(5)then(6)Finally, if(7)then(8)
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)
The integral of over the unit disk is given by(1)(2)(3)(4)In general,(5)provided .Additional integrals include(6)(7)(8)
A number defined by , where is a Bernoulli polynomial of the second kind (Roman 1984, p. 294), also called Cauchy numbers of the first kind. The first few for , 1, 2, ... are 1, 1/2, , 1/4, , 9/4, ... (OEIS A006232 and A006233). They are given bywhere is a falling factorial, and have exponential generating function
The Poisson integral with ,where is a Bessel function of the first kind.
Special cases of general formulas due to Bessel.where is a Bessel function of the first kind. Now, let and . Then
The function defined by the contour integralwhere denotes the contour encircling the point once in a counterclockwise direction. It is equal to(Watson 1966, p. 326).
The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let be an angle measured counterclockwise from the x-axis along the arc of the unit circle. Then is the horizontal coordinate of the arc endpoint.The common schoolbook definition of the cosine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle adjacent to the angle and the hypotenuse, i.e.,(1)A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).As a result of its definition, the cosine function is periodic with period . By the Pythagorean theorem, also obeys the identity(2) Min Max Re Im The definition of the cosine function can be extended to..
A Bessel function of the second kind (e.g, Gradshteyn and Ryzhik 2000, p. 703, eqn. 6.649.1), sometimes also denoted (e.g, Gradshteyn and Ryzhik 2000, p. 657, eqn. 6.518), is a solution to the Bessel differential equation which is singular at the origin. Bessel functions of the second kind are also called Neumann functions or Weber functions. The above plot shows for , 1, 2, ..., 5. The Bessel function of the second kind is implemented in the Wolfram Language as BesselY[nu, z].Let be the first solution and be the other one (since the Bessel differential equation is second-order, there are two linearly independent solutions). Then(1)(2)Take (1) minus (2),(3)(4)so , where is a constant. Divide by ,(5)(6)Rearranging and using gives(7)(8)where is the so-called Bessel function of the second kind. can be defined by(9)(Abramowitz and Stegun 1972, p. 358), where is a Bessel function of the first kind and, for an integer..
The sine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cosine, cotangent, secant, and tangent). Let be an angle measured counterclockwise from the x-axis along an arc of the unit circle. Then is the vertical coordinate of the arc endpoint, as illustrated in the left figure above.The common schoolbook definition of the sine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle opposite the angle and the hypotenuse, i.e.,(1)A convenient mnemonic for remembering the definition of the sine, as well as the cosine and tangent, is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).As a result of its definition, the sine function is periodic with period . By the Pythagorean theorem, also obeys the identity(2) Min Max Re Im The definition..
The Bessel functions of the first kind are defined as the solutions to the Bessel differential equation(1)which are nonsingular at the origin. They are sometimes also called cylinder functions or cylindrical harmonics. The above plot shows for , 1, 2, ..., 5. The notation was first used by Hansen (1843) and subsequently by Schlömilch (1857) to denote what is now written (Watson 1966, p. 14). However, Hansen's definition of the function itself in terms of the generating function(2)is the same as the modern one (Watson 1966, p. 14). Bessel used the notation to denote what is now called the Bessel function of the first kind (Cajori 1993, vol. 2, p. 279).The Bessel function can also be defined by the contour integral(3)where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).The Bessel function of the first kind is implemented in the Wolfram Language as BesselJ[nu,..
An entire function which is a generalization of the Bessel function of the first kind defined byAnger's original function had an upper limit of , but the current notation was standardized by Watson (1966).The Anger function may also be written aswhere is a regularized hypergeometric function.If is an integer , then , where is a Bessel function of the first kind.The Anger function is implemented in the Wolfram Language as AngerJ[nu, z].
Apéry's constant is defined by(1)(OEIS A002117) where is the Riemann zeta function. Apéry (1979) proved that is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of (Hata 2000). arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.The following table summarizes progress in computing upper bounds on the irrationality measure for . Here, the exact values for is given by(2)(3)(Hata 2000).upper boundreference15.513891Rhin and Viola (2001)28.830284Hata (1990)312.74359Dvornicich and Viola (1987)413.41782Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)Beukers (1979) reproduced Apéry's rational approximation to using the triple..
Graph the Riemann sum of as x goes from to using rectangles taking samples at the Maximum Minimum Left Right Midpoint Print estimated and actual areas? Rectangle Color Plot Color Light Gray Dark Gray Black White Red Orange Yellow Green Blue Purple Light Gray Dark Gray Black White Red Orange Yellow Green Blue Purple Replot Let a closed interval be partitioned by points , where the lengths of the resulting intervals between the points are denoted , , ..., . Let be an arbitrary point in the th subinterval. Then the quantityis called a Riemann sum for a given function and partition, and the value is called the mesh size of the partition.If the limit of the Riemann sums exists as , this limit is known as the Riemann integral of over the interval . The shaded areas in the above plots show the lower and upper sums for a constant mesh size...
Min Max Re Im A special function which is given by the logarithmic derivative of the gamma function (or, depending on the definition, the logarithmic derivative of the factorial).Because of this ambiguity, two different notations are sometimes (but not always) used, with(1)defined as the logarithmic derivative of the gamma function , and(2)defined as the logarithmic derivative of the factorial function. The two are connected by the relationship(3)The th derivative of is called the polygamma function, denoted . The notation(4)is therefore frequently used for the digamma function itself, and Erdélyi et al. (1981) use the notation for . The digamma function is returned by the function PolyGamma[z] or PolyGamma[0, z] in the Wolfram Language, and typeset using the notation .The digamma function arises in simple sums such as(5)(6)where is a Lerch transcendent.Special cases are given by(7)(8)(9)(10)Gauss's digamma theorem states..
Min Max Re Im The central beta function is defined by(1)where is the beta function. It satisfies the identities(2)(3)(4)(5)With , the latter gives the Wallis formula. For , 2, ... the first few values are 1, 1/6, 1/30, 1/140, 1/630, 1/2772, ... (OEIS A002457), which have denominators .When ,(6)where(7)The central beta function satisfies(8)(9)(10)(11)For an odd positive integer, the central beta function satisfies the identity(12)
For ,where is the gamma function.
Binet's first formula for , where is a gamma function, is given byfor (Erdélyi et al. 1981, p. 21; Whittaker and Watson 1990, p. 249).Binet's second formula isfor (Erdélyi et al. 1981, p. 22; Whittaker and Watson 1990, pp. 250-251).
Hadjicostas's formula is a generalization of the unitsquare double integral(1)(Sondow 2003, 2005; Borwein et al. 2004, p. 49), where is the Euler-Mascheroni constant. It states(2)for , where is the gamma function and is the Riemann zeta function (although care must be taken at because of the removable singularity present there). It was conjectured by Hadjicostas (2004) and almost immediately proved by Chapman (2004). The special case gives Beukers's integral for ,(3)(Beukers 1979). At , the formula is related to Beukers's integral for Apéry's constant , which is how interest in this class of integrals originally arose.There is an analogous formula(4)for , due to Sondow (2005), where is the Dirichlet eta function. This includes the special cases(5)(6)(7)(OEIS A094640; Sondow 2005) and(8)(9)(OEIS A103130), where is the Glaisher-Kinkelin constant (Sondow 2005)...
The beta function is the name used by Legendre and Whittaker and Watson (1990) for the beta integral (also called the Eulerian integral of the first kind). It is defined by(1)The beta function is implemented in the Wolfram Language as Beta[a, b].To derive the integral representation of the beta function, write the product oftwo factorials as(2)Now, let , , so(3)(4)Transforming to polar coordinates with , (5)(6)(7)(8)The beta function is then defined by(9)(10)Rewriting the arguments then gives the usual form for the beta function,(11)(12)By symmetry,(13)The general trigonometric form is(14)Equation (14) can be transformed to an integral over polynomials by letting ,(15)(16)(17)(18)For any with ,(19)(Krantz 1999, p. 158).To put it in a form which can be used to derive the Legendre duplication formula, let , so and , and(20)(21)To put it in a form which can be used to develop integral representations of the Bessel functions and hypergeometric..
Min Max Min Max Re Im The plots above show the values of the function obtained by taking the natural logarithm of the gamma function, . Note that this introduces complicated branch cut structure inherited from the logarithm function. Min Max Re Im For this reason, the logarithm of the gamma function is sometimes treated as a special function in its own right, and defined differently from . This special "log gamma" function is implemented in the Wolfram Language as LogGamma[z], plotted above. As can be seen, the two definitions have identical real parts, but differ markedly in their imaginary components. Most importantly, although the log gamma function and are equivalent as analytic multivalued functions, they have different branch cut structures and a different principal branch, and the log gamma function is analytic throughout the complex -plane except for a single branch cut discontinuity along the negative real axis. In particular,..
(1)(2)It is equivalent to(3)where is the En-function.
Min Max The absolute value of a real number is denoted and defined as the "unsigned" portion of ,(1)(2)where is the sign function. The absolute value is therefore always greater than or equal to 0. The absolute value of for real is plotted above. Min Max Re Im The absolute value of a complex number , also called the complex modulus, is defined as(3)This form is implemented in the Wolfram Language as Abs[z] and is illustrated above for complex .Note that the derivative (read: complex derivative) does not exist because at every point in the complex plane, the value of the derivative of depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than as(4)As a result of the fact that computer algebra languages such as the Wolfram Language generically deal with..
Borwein et al. (2004, pp. 4 and 44) term the expression of the integrals(1)(2)(3)(4)(OEIS A083648 and A073009)in terms of infinite sums "a sophomore's dream."For , write(5)(6)Integrating term by term then gives(7)(8)(9)(Borwein et al. 2004, p. 44).For , write(10)(11)Integrating term by term then gives(12)(13)(14)(Borwein et al. 2004, pp. 4 and 44).
Min Max Re Im The cotangent function is the function defined by(1)(2)(3)where is the tangent. The cotangent is implemented in the Wolfram Language as Cot[z].The notations (Erdélyi et al. 1981, p. 7; Jeffrey 2000, p. 111) and (Gradshteyn and Ryzhik 2000, p. xxix) are sometimes used in place of . Note that the cotangent is not in as widespread use in Europe as are , , and , although it does appear explicitly in various German and Russian handbooks (e.g., Gradshteyn and 2000, p. 28). Interestingly, is treated on par with the other trigonometric functions in most tabulations (Gellert et al. 1989, p. 222; Gradshteyn and Ryzhik 2000, p. 28), while and are sometimes not (Gradshteyn and Ryzhik 2000, p. 28).An important identity connecting the cotangent with the cosecantis given by(4)The cotangent has smallest real fixed point such at 0.8603335890... (OEIS A069855; Bertrand 1865, p. 285).The..
A formal extension of the hypergeometric function to two variables, resulting in four kinds of functions (Appell 1925; Picard 1880ab, 1881; Goursat 1882; Whittaker and Watson 1990, Ex. 22, p. 300),(1)(2)(3)(4)These double series are absolutely convergent for(5)Appell defined the functions in 1880 and they were subsequently studied by Picard in 1881. The functions , , and can be expressed in terms of double integrals as(6)(7)(8)(Bailey 1934, pp. 76-77). There appears to be no simple integral representation of this type for the function (Bailey 1934, p. 77).The function can also be expressed by the simple integral(9)(Bailey 1934, p. 77), for and .The Appell functions are special cases of the Kampé de Fériet function, and are the first four in the set of Horn functions. The function is implemented in the Wolfram Language as AppellF1[a, b1, b2, c, x, y].For general complex parameters, the function..
The complete elliptic integral of the second kind, illustrated above as a function of , is defined by(1)(2)(3)(4)where is an incomplete elliptic integral of the second kind, is the hypergeometric function, and is a Jacobi elliptic function.It is implemented in the Wolfram Language as EllipticE[m], where is the parameter. can be computed in closed form in terms of and the elliptic alpha function for special values of , where is a called an elliptic integral singular value. Other special values include(5)(6)The complete elliptic integral of the second kind satisfies the Legendrerelation(7)where and are complete elliptic integrals of the first and second kinds, respectively, and and are the complementary integrals. The derivative is(8)(Whittaker and Watson 1990, p. 521).The solution to the differential equation(9)(Zwillinger 1997, p. 122; Gradshteyn and Ryzhik 2000, p. 907) is given by(10)If is a singular value..
The complete elliptic integral of the first kind , illustrated above as a function of the elliptic modulus , is defined by(1)(2)(3)where is the incomplete elliptic integral of the first kind and is the hypergeometric function.It is implemented in the Wolfram Language as EllipticK[m], where is the parameter.It satisfies the identity(4)where is a Legendre polynomial. This simplifies to(5)for all complex values of except possibly for real with .In addition, satisfies the identity(6)where is the complementary modulus. Amazingly, this reduces to the beautiful form(7)for (Watson 1908, 1939). can be computed in closed form for special values of , where is a called an elliptic integral singular value. Other special values include(8)(9)(10)(11)(12) satisfies(13)possibly modulo issues of , which can be derived from equation 17.4.17 in Abramowitz and Stegun (1972, p. 593). is related to the Jacobi elliptic functions through(14)where..
The line integral of a vector field on a curve is defined by(1)where denotes a dot product. In Cartesian coordinates, the line integral can be written(2)where(3)For complex and a path in the complex plane parameterized by ,(4)Poincaré's theorem states that if in a simply connected neighborhood of a point , then in this neighborhood, is the gradient of a scalar field ,(5)for , where is the gradient operator. Consequently, the gradient theorem gives(6)for any path located completely within , starting at and ending at .This means that if (i.e., is an irrotational field in some region), then the line integral is path-independent in this region. If desired, a Cartesian path can therefore be chosen between starting and ending point to give(7)If (i.e., is a divergenceless field, a.k.a. solenoidal field), then there exists a vector field such that(8)where is uniquely determined up to a gradient field (and which can be chosen so that )...
There are four varieties of Airy functions: , , , and . Of these, and are by far the most common, with and being encountered much less frequently. Airy functions commonly appear in physics, especially in optics, quantum mechanics, electromagnetics, and radiative transfer. and are entire functions.A generalization of the Airy function was constructed by Hardy.The Airy function and functions are plotted above along the real axis.The and functions are defined as the two linearly independent solutions to(1)(Abramowitz and Stegun 1972, pp. 446-447; illustrated above), written in the form(2)where(3)(4)where is a confluent hypergeometric limit function. These functions are implemented in the Wolfram Language as AiryAi[z] and AiryBi[z]. Their derivatives are implemented as AiryAiPrime[z] and AiryBiPrime[z].For the special case , the functions can be written as(5)(6)(7)where is a modified Bessel function of the first kind and..
The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in , and states that if is defined by the integral (antiderivative)thenat each point in , where is the derivative of .
The Leibniz integral rule gives a formula for differentiation of a definiteintegral whose limits are functions of the differential variable,(1)It is sometimes known as differentiation under the integral sign.This rule can be used to evaluate certain unusual definite integrals such as(2)(3)for (Woods 1926).Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."
The first fundamental theorem of calculus states that, if is continuous on the closed interval and is the indefinite integral of on , then(1)This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.The second fundamental theorem of calculus holds for a continuous function on an open interval and any point in , and states that if is defined by(2)then(3)at each point in .The fundamental theorem of calculus along curves states that if has a continuous indefinite integral in a region containing a parameterized curve for , then(4)
The bivariate normal distribution is the statistical distribution with probabilitydensity function(1)where(2)and(3)is the correlation of and (Kenney and Keeping 1951, pp. 92 and 202-205; Whittaker and Robinson 1967, p. 329) and is the covariance.The probability density function of the bivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, sigma11, sigma12, sigma12, sigma22] in the Wolfram Language package MultivariateStatistics` .The marginal probabilities are then(4)(5)and(6)(7)(Kenney and Keeping 1951, p. 202).Let and be two independent normal variates with means and for , 2. Then the variables and defined below are normal bivariates with unit variance and correlation coefficient :(8)(9)To derive the bivariate normal probability function, let and be normally and independently distributed variates with mean 0 and variance 1, then define(10)(11)(Kenney and Keeping..
A definite integral is an integral(1)with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral (which is the usual definition encountered in elementary textbooks). However, a general definite integral is taken in the complex plane, resulting in the contour integral(2)with , , and in general being complex numbers and the path of integration from to known as a contour.The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if is the indefinite integral for a continuous function , then(3)This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Definite integrals may be evaluated in the Wolfram Language using Integrate[f, x, a, b].The question of which definite..
Catalan's constant is a constant that commonly appears in estimates of combinatorial functions and in certain classes of sums and definite integrals. It is usually denoted (this work), (e.g., Borwein et al. 2004, p. 49), or (Wolfram Language).Catalan's constant may be defined by(1)(Glaisher 1877, who however did not explicitly identify the constant in this paper). It is not known if is irrational.Catalan's constant is implemented in the WolframLanguage as Catalan.The constant is named in honor of E. C. Catalan (1814-1894), who first gave an equivalent series and expressions in terms of integrals. Numerically,(2)(OEIS A006752). can be given analytically by the following expressions(3)(4)(5)where is the Dirichlet beta function, is Legendre's chi-function, is the Glaisher-Kinkelin constant, and is the partial derivative of the Hurwitz zeta function with respect to the first argument.Glaisher (1913) gave(6)(Vardi..
There are many formulas of of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. is intimately related to the properties of circles and spheres. For a circle of radius , the circumference and area are given by(1)(2)Similarly, for a sphere of radius , the surface area and volume enclosed are(3)(4)An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula(5)There are three other Machin-like formulas,as well as thousands of other similar formulas having more terms.Gregory and Leibniz found(6)(7)(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The error after the th term of this series in the Gregory series is larger than so this sum converges so slowly that 300 terms are not sufficient to calculate correctly to two decimal places! However, it can be transformed..
The figure eight knot, also known as the Flemish knot and savoy knot, is the unique prime knot of four crossings 04-001. It has braid word .The figure eight knot is implemented in the WolframLanguage as KnotData["FigureEight"].It is a 2-embeddable knot, and is amphichiral as well as invertible. It has Arf invariant 1. It is not a slice knot (Rolfsen 1976, p. 224).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the figure eight knot are(1)(2)(3)(4)(5)(6)There are no other knots on 10 or fewer crossings sharing the same Alexander polynomial, BLM/Ho polynomial, bracket polynomial, HOMFLY polynomial, Jones polynomial, or Kauffman polynomial F.The figure eight knot has knot group(7)(Rolfsen 1976, p. 58).Helaman Ferguson's sculpture "Figure-Eight Complement II" illustrates the knot complement of the figure eight..
The Lerch transcendent is generalization of the Hurwitz zeta function and polylogarithm function. Many sums of reciprocal powers can be expressed in terms of it. It is classically defined by(1)for and , , .... It is implemented in this form as HurwitzLerchPhi[z, s, a] in the Wolfram Language.The slightly different form(2)sometimes also denoted , for (or and ) and , , , ..., is implemented in the Wolfram Language as LerchPhi[z, s, a]. Note that the two are identical only for .A series formula for valid on a larger domain in the complex -plane is given by(3)which holds for all complex and complex with (Guillera and Sondow 2005).The Lerch transcendent can be used to express the Dirichletbeta function(4)(5)A special case is given by(6)(Guillera and Sondow 2005), where is the polylogarithm.Special cases giving simple constants include(7)(8)(9)(10)where is Catalan's constant, is Apéry's constant, and is the Glaisher-Kinkelin constant..
Define the Airy zeta function for , 3, ... by(1)where the sum is over the real (negative) zeros of the Airy function . This has the closed-form representation(2)where is the gamma function,(3)where(4)(5)and(6)(Crandall 1996; Borwein et al. 2004, p. 61).Surprisingly, defining(7)(8)(9)gives as a polynomial in (Borwein et al. 2004, pp. 61-62). The first few such polynomials are(10)(11)(12)(13)(14)(OEIS A096631 and A096632). The corresponding numerical values are approximately 0.531457, , 0.0394431, , and 0.00638927, ....
A multiple integral is a set of integrals taken over variables, e.g.,An th-order integral corresponds, in general, to an -dimensional volume (i.e., a content), with corresponding to an area. In an indefinite multiple integral, the order in which the integrals are carried out can be varied at will; for definite multiple integrals, care must be taken to correctly transform the limits if the order is changed.In traditional mathematical notation, a multiple integral of a function that is first performed over a variable and then performed over a variable is writtenIn the Wolfram Language, this would be entered as Integrate[f[x, y], x, x1, x2, y, y1[x], y2[x]], where the order of the integration variables is specified in the order that the integral signs appear on the left, which is opposite to the actual order of integration...
The first fundamental theorem of calculus states that, if is continuous on the closed interval and is the indefinite integral of on , thenThis result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.
The Cantor function is the continuous but not absolutely continuous function on which may be defined as follows. First, express in ternary. If the resulting ternary digit string contains the digit 1, replace every ternary digit following the 1 by a 0. Next, replace all 2's with 1's. Finally, interpret the result as a binary number which then gives .The Cantor function is a particular case of a devil's staircase (Devaney 1987, p. 110), and can be extended to a function for , with corresponding to the usual Cantor function (Gorin and Kukushkin 2004).Chalice (1991) showed that any real-valued function on which is monotone increasing and satisfies 1. , 2. , 3. is the Cantor function (Chalice 1991; Wagon 2000, p. 132).Gorin and Kukushkin (2004) give the remarkable identityfor integer . For and , 2, ..., this gives the first few values as 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, ... (OEIS A095844 and A095845).M. Trott (pers. comm., June..
The Borwein integrals are the class of definiteintegrals defined byfor odd . The integrals are curious because the terms , 3, ..., 13 all have unit numerators, but , 17, ... do not. The sequence of values of for , 3, ... is given by 1/2, 1/6, 1/30, 1/210, 1/1890, 1/20790, 1/270270, 467807924713440738696537864469/1896516717212415135141110350293750000, ... (OEIS A068214 and A068215; Borwein et al. 2004, p. 98; Bailey et al. 2006).
Erfc is the complementary error function, commonly denoted , is an entire function defined by(1)(2)It is implemented in the Wolfram Languageas Erfc[z].Note that some authors (e.g., Whittaker and Watson 1990, p. 341) define without the leading factor of .For ,(3)where is the incomplete gamma function.The derivative is given by(4)and the indefinite integral by(5)It has the special values(6)(7)(8)It satisfies the identity(9)It has definite integrals(10)(11)(12)For , is bounded by(13) Min Max Re Im Erfc can also be extended to the complex plane, as illustrated above.A generalization is obtained from the erfcdifferential equation(14)(Abramowitz and Stegun 1972, p. 299; Zwillinger 1997, p. 122). The general solution is then(15)where is the repeated erfc integral. For integer ,(16)(17)(18)(19)(Abramowitz and Stegun 1972, p. 299), where is a confluent hypergeometric function of the first kind and is a..
Min Max Min Max Re Im The Dirichlet eta function is the function defined by(1)(2)where is the Riemann zeta function. Note that Borwein and Borwein (1987, p. 289) use the notation instead of . The function is also known as the alternating zeta function and denoted (Sondow 2003, 2005). is defined by setting in the right-hand side of (2), while (sometimes called the alternating harmonic series) is defined using the left-hand side. The function vanishes at each zero of except (Sondow 2003).The eta function is related to the Riemann zeta function and Dirichlet lambda function by(3)and(4)(Spanier and Oldham 1987). The eta function is also a special case of the polylogarithmfunction,(5)The value may be computed by noting that the Maclaurin series for for is(6)Therefore, the natural logarithm of 2 is(7)(8)(9)(10)The derivative of the eta function is given by(11)or in the special case , by(12)(13)(14)(15)This latter fact provides a remarkable..
Min Max Min Max Re Im The Dirichlet beta function is defined by the sum(1)(2)where is the Lerch transcendent. The beta function can be written in terms of the Hurwitz zeta function by(3)The beta function can be defined over the whole complexplane using analytic continuation,(4)where is the gamma function.The Dirichlet beta function is implemented in the WolframLanguage as DirichletBeta[x].The beta function can be evaluated directly special forms of arguments as(5)(6)(7)where is an Euler number.Particular values for are(8)(9)(10)(11)where is Catalan's constant and is the polygamma function. For , 3, 5, ..., , where the multiples are 1/4, 1/32, 5/1536, 61/184320, ... (OEIS A046976 and A053005).It is involved in the integral(12)(Guillera and Sondow 2005).Rivoal and Zudilin (2003) proved that at least one of the seven numbers , , , , , , and is irrational.The derivative can also be computed analytically at a number of integer values of including(13)(14)(15)(16)(17)(18)(19)(OEIS..
The inverse erf function is the inverse function of such that(1)with the first identity holding for and the second for . It is implemented in the Wolfram Language as InverseErfc[z].It is related to inverse erf by(2)It has the special values(3)(4)(5)It has the derivative(6)and its indefinite integral is(7)(which follows from the method of Parker 1955).The Taylor series about 1 is given by(8)(OEIS A002067 and A007019).
The inverse erf function is the inverse function of the erf function such that(1)(2)with the first identity holding for and the second for . It is implemented in the Wolfram Language as InverseErf[x].It is an odd function since(3)It has the special values(4)(5)(6)It is apparently not known if(7)(OEIS A069286) can be written in closed form.It satisfies the equation(8)where is the inverse erfc function.It has the derivative(9)and its integral is(10)(which follows from the method of Parker 1955).Definite integrals are given by(11)(12)(13)(14)(OEIS A087197 and A114864), where is the Euler-Mascheroni constant and is the natural logarithm of 2.The Maclaurin series of is given by(15)(OEIS A002067 and A007019). Written in simplified form so that the coefficient of is 1,(16)(OEIS A092676 and A092677). The th coefficient of this series can be computed as(17)where is given by the recurrence equation(18)with initial condition ...
The largest value of a set, function, etc. The maximum value of a set of elements is denoted or , and is equal to the last element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the maximum is 5. The maximum and minimum are the simplest order statistics.The maximum value of a variable is commonly denoted (Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 74). In this work, the convention is used.The maximum of a set of elements is implemented in the Wolfram Language as Max[list] and satisfies the identities(1)(2)Definite integrals include(3)(4)A continuous function may assume a maximum at a single point or may have maxima at a number of points. A global maximum of a function is the largest value in the entire range of the function, and a local maximum is the largest value in some local neighborhood.For a function which is continuous at a point , a necessary but not sufficient condition..
The Cauchy principal value of a finite integral of a function about a point with is given by(Henrici 1988, p. 261; Whittaker and Watson 1990, p. 117; Bronshtein and Semendyayev 1997, p. 283). Similarly, the Cauchy principal value of a doubly infinite integral of a function is defined byThe Cauchy principal value is also known as the principal value integral (Henrici 1988, p. 261), finite part (Vladimirov 1971), or partie finie (Vladimirov 1971).The Cauchy principal value of an integral having no nonsimple poles can be computed in the Wolfram Language using Integrate[f, x, a, b, PrincipalValue -> True]. Cauchy principal values of functions with possibly nonsimple poles can be computed numerically using the "CauchyPrincipalValue" method in NIntegrate.Cauchy principal values are important in the theory of generalized functions, where they allow extension of results to .Cauchy principal values..
Let be a real continuous monotonic strictly increasing function on the interval with and , thenwhere is the inverse function. Equality holds iff .
At the age of 17, Bernard Mares proposed the definite integral (Borwein and Bailey2003, p. 26; Bailey et al. 2006)(1)(2)(OEIS A091473). Although this is within of ,(3)(OEIS A091494), it is not equal to it. Apparently, no closed-form solution is known for .Interestingly, the integral(4)(5)(Borwein et al. 2004, pp. 101-102) has a value fairly close to , but no other similar relationships seem to hold for other multipliers of the form or .The identity(6)can be expanded to yield(7)In fact,(8)where is a Borwein integral.