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### Reuleaux tetrahedron

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..

### Square point picking

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..

### Hypercube line picking

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..

### Method of washers

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

### Method of shells

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

### Method of disks

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

### Triangle line picking

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..

### Poincar&eacute;'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.

### Smooth curve

A smooth curve is a curve which is a smooth function, where the word "curve" is interpreted in the analytic geometry context. In particular, a smooth curve is a continuous map from a one-dimensional space to an -dimensional space which on its domain has continuous derivatives up to a desired order.

### Vector integral

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)

### Normal vector

The normal vector, often simply called the "normal," to a surface is a vector which is perpendicular to the surface at a given point. When normals are considered on closed surfaces, the inward-pointing normal (pointing towards the interior of the surface) and outward-pointing normal are usually distinguished.The unit vector obtained by normalizing the normal vector (i.e., dividing a nonzero normal vector by its vector norm) is the unit normal vector, often known simply as the "unit normal." Care should be taken to not confuse the terms "vector norm" (length of vector), "normal vector" (perpendicular vector) and "normalized vector" (unit-length vector).The normal vector is commonly denoted or , with a hat sometimes (but not always) added (i.e., and ) to explicitly indicate a unit normal vector.The normal vector at a point on a surface is given by(1)where and are partial derivatives.A..

### 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)

### Bernoulli number of the second kind

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

### Catalan integrals

Special cases of general formulas due to Bessel.where is a Bessel function of the first kind. Now, let and . Then

### Bourget function

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).

### Cosine

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..

### Bessel function of the second kind

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..

### Sine

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..

### Bessel function of the first kind

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,..

### Anger function

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].

### Ap&eacute;ry's constant

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..

### Jacobian

Given a set of equations in variables , ..., , written explicitly as(1)or more explicitly as(2)the Jacobian matrix, sometimes simply called "the Jacobian" (Simon and Blume 1994) is defined by(3)The determinant of is the Jacobian determinant (confusingly, often called "the Jacobian" as well) and is denoted(4)The Jacobian matrix and determinant can be computed in the WolframLanguage using JacobianMatrix[f_List?VectorQ, x_List] := Outer[D, f, x] /; Equal @@ (Dimensions /@ {f, x}) JacobianDeterminant[f_List?VectorQ, x_List] := Det[JacobianMatrix[f, x]] /; Equal @@ (Dimensions /@ {f, x})Taking the differential(5)shows that is the determinant of the matrix , and therefore gives the ratios of -dimensional volumes (contents) in and ,(6)It therefore appears, for example, in the changeof variables theorem.The concept of the Jacobian can also be applied to functions in more than variables. For example, considering..

### Green's theorem

Green's theorem is a vector identity which is equivalent to the curl theorem in the plane. Over a region in the plane with boundary , Green's theorem states(1)where the left side is a line integral and the right side is a surface integral. This can also be written compactly in vector form as(2)If the region is on the left when traveling around , then area of can be computed using the elegant formula(3)giving a surprising connection between the area of a region and the line integral around its boundary. For a plane curve specified parametrically as for , equation (3) becomes(4)which gives the signed area enclosed by the curve.The symmetric for above corresponds to Green's theorem with and , leading to(5)(6)(7)(8)(9)However, we are also free to choose other values of and , including and , giving the "simpler" form(10)and and , giving(11)A similar procedure can be applied to compute the moment about the -axis using and as(12)and about the..

### Digamma function

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..

### Central beta function

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)

### Binet's log gamma formulas

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).

### Beta function

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..

### Log gamma function

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,..

### Alpha function

(1)(2)It is equivalent to(3)where is the En-function.

### Lower bound

A function is said to have a lower bound if for all in its domain. The greatest lower bound is called the infimum.

### Upper bound

A function is said to have a upper bound if for all in its domain. The least upper bound is called the supremum. A set is said to be bounded from above if it has an upper bound.

### Absolute value

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..

### Sophomore's dream

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).

### Cotangent

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..

### Appell hypergeometric function

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..

### Complete elliptic integral of the second kind

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..

### Complete elliptic integral of the first kind

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..

### Line integral

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 )...

### Airy functions

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..

### Bivariate normal distribution

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..

### Catalan's constant

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..

### Pi formulas

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..

### Figure eight knot

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..

### Lerch transcendent

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..

### Airy zeta function

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, ....

### Tangent vector

For a curve with radius vector , the unit tangent vector is defined by(1)(2)(3)where is a parameterization variable, is the arc length, and an overdot denotes a derivative with respect to , . For a function given parametrically by , the tangent vector relative to the point is therefore given by(4)(5)To actually place the vector tangent to the curve, it must be displaced by . It is also true that(6)(7)(8)where is the normal vector, is the curvature, is the torsion, and is the scalar triple product.

### Curvature

In general, there are two important types of curvature: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two- and three-space was the first type of curvature to be studied historically, culminating in the Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and the initial starting point and direction.After the curvature of two- and three-dimensional curves was studied, attention turned to the curvature of surfaces in three-space. The main curvatures that emerged from this scrutiny are the mean curvature, Gaussian curvature, and the shape operator. Mean curvature was the most important for applications at the time and was the most studied, but Gauss was the first to recognize the importance of the Gaussian curvature.Because Gaussian curvature is "intrinsic," it is detectable to two-dimensional "inhabitants" of the surface,..

### Arc length

Arc length is defined as the length along a curve,(1)where is a differential displacement vector along a curve . For example, for a circle of radius , the arc length between two points with angles and (measured in radians) is simply(2)Defining the line element , parameterizing the curve in terms of a parameter , and noting that is simply the magnitude of the velocity with which the end of the radius vector moves gives(3)In polar coordinates,(4)so(5)(6)In Cartesian coordinates,(7)(8)(9)(10)Therefore, if the curve is written(11)then(12)If the curve is instead written(13)then(14)In three dimensions,(15)so(16)The arc length of the polar curve is given by(17)

### Three circles theorem

The three circles theorem, also called Hadamard's three circles theorem (Edwards 2001, p. 187), states that if is an analytic function in the annulus , , and , , and are the maxima of on the three circles corresponding to , , and , respectively, then(Derbyshire 2004, p. 376).The theorem was first published by Hadamard in 1896, although without proof (Bohr and Landau 1913; Edwards 2001, p. 187).

### Inflection point

An inflection point is a point on a curve at which the sign of the curvature (i.e., the concavity) changes. Inflection points may be stationary points, but are not local maxima or local minima. For example, for the curve plotted above, the point is an inflection point.The first derivative test can sometimes distinguish inflection points from extrema for differentiable functions .The second derivative test is also useful. A necessary condition for to be an inflection point is . A sufficient condition requires and to have opposite signs in the neighborhood of (Bronshtein and Semendyayev 2004, p. 231).

The conjugate gradient method is an algorithm for finding the nearest local minimum of a function of variables which presupposes that the gradient of the function can be computed. It uses conjugate directions instead of the local gradient for going downhill. If the vicinity of the minimum has the shape of a long, narrow valley, the minimum is reached in far fewer steps than would be the case using the method of steepest descent.For a discussion of the conjugate gradient method on vector and shared memory computers, see Dongarra et al. (1991). For discussions of the method for more general parallel architectures, see Demmel et al. (1993) and Ortega (1988) and the references therein.

### Cantor function

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..

### Borwein integrals

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

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..

### Dirichlet eta function

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..

### Dirichlet beta function

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..

### Inverse erfc

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).

### Inverse erf

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 ...

### Young's integral

Let be a real continuous monotonic strictly increasing function on the interval with and , thenwhere is the inverse function. Equality holds iff .

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