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Special Functions

This worksheet gives definitions, properties, and graphs of some of the special mathematical functions available in Maple.

>	$Typesetting :- EnableTypesetRule (Typesetting :- SpecialFunctionRules) &colon;$

The Airy Wave Functions

The Airy Ai and Bi functions, AiryAi(z) and AiryBi(z) respectively, solve the differential equation:
$\frac{\partial^{2}}{\partial z \partial z} y (z) - z y (z) = 0$

Floating-point Evaluation

>	$AiryAi (3.2)$

$0.004567439274$

(1.1.1)

Differentiation

The derivatives of the Airy functions are denoted by AiryAi(1, z) and AiryBi(1, z):

>	$\frac{&DifferentialD;}{&DifferentialD; z} AiryAi (z), \frac{&DifferentialD;}{&DifferentialD; z} AiryBi (z)$

$Ai' (z), Bi' (z)$

(1.2.1)

>	$\frac{&DifferentialD;}{&DifferentialD; z} AiryAi (1, z), \frac{&DifferentialD;}{&DifferentialD; z} AiryBi (1, z)$

$z Ai (z), z Bi (z)$

(1.2.2)

Graphing

>	$plot ([AiryAi (z), AiryBi (z)], z = - 3 .. 1, legend = [Ai, Bi])$

>	$plot ([seq (diff (AiryAi (z), [z &dollar; i]), i = 0 .. 4)], z = - 3 .. 1, legend = [Ai, 1st Derivative, 2nd Derivative, 3rd Derivative, 4th Derivative])$

The Anger and Weber Functions

The Anger J function, AngerJ(nu, z), solves the inhomogeneous Bessel equation:
$z^{2} (\frac{\partial^{2}}{\partial z \partial z} y (z)) + z (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) + (z^{2} - ν^{2}) y (z) = \frac{(z - ν) \sin (π ν)}{π}$

and the Weber E function, WeberE(nu, z), solves the inhomogeneous Bessel equation:
$z^{2} (\frac{\partial^{2}}{\partial z \partial z} y (z)) + z (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) + (z^{2} - ν^{2}) y (z) = \frac{(ν - z) \cos (π ν) - (ν + z)}{π}$

Floating-point Evaluation

>	$AngerJ (2.0, 3.2), BesselJ (2.0, 3.2)$

$0.4835277001, 0.4835277001$

(2.1.1)

>	$AngerJ (0.5, 3.2), BesselJ (0.5, 3.2)$

$0.05605486408, −0.02603667926$

(2.1.2)

>	$WeberE (0.0, 3.2), - StruveH (0.0, 3.2)$

$−0.4933957395, −0.4933957395$

(2.1.3)

>	$WeberE (1.0, 0.5), evalf (\frac{2}{π}) - StruveH (1.0, 0.5)$

$0.5844460281, 0.5844460280$

(2.1.4)

Differentiation

The derivatives with respect to the variable can be expressed in terms of algebraic expressions in trigonometric functions and the Anger and Weber functions themselves with shifted parameters:

>	$diff (AngerJ (nu, z), z)$

$- J_{ν + 1} (z) + \frac{ν J_{ν} (z)}{z} - \frac{\sin (ν π)}{π z}$

(2.2.1)

>	$diff (WeberE (nu, z), z)$

$- E_{ν + 1} (z) + \frac{ν E_{ν} (z)}{z} - \frac{1 - \cos (ν π)}{π z}$

(2.2.2)

Graphing

>	$plot ([AngerJ (2.5, z), WeberE (2.5, z)], z = 0 .. 10, legend = [J, E])$

>	$plot ([AngerJ (2.5, z), \frac{&DifferentialD;}{&DifferentialD; z} AngerJ (2.5, z)], z = 0 .. 10, legend = [J, 1st Derivative])$

>	$plot ([seq (AngerJ (\frac{i}{2}, z), i = 0 .. 4)], z = 0 .. 10, legend = [J 0, J 1/2, J 1, J 3/2, J 2])$

The Bessel Functions

The Bessel J and Y functions of the first kind, BesselJ(nu, z) and BesselY(nu,z), satisfy the differential equation:
$z^{2} (\frac{\partial^{2}}{\partial z \partial z} y (z)) + z (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) + (z^{2} - ν^{2}) y (z) = 0$

The Bessel I and K functions of the second kind, BesselI(nu,z) and BesselK(nu,z), satisfy the modified differential equation:
$z^{2} (\frac{\partial^{2}}{\partial z \partial z} y (z)) + z (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) - (z^{2} + ν^{2}) y (z) = 0$

The Hankel $H^{(1)}$ and $H^{(2)}$ functions, HankelH1(nu,z) and HankelH2(nu,z) are related to the Bessel functions of the first kind by:
$HankelH1 (ν, z) = BesselJ (ν, z) + I BesselY (ν, z)$
$HankelH2 (ν, z) = BesselJ (ν, z) - I BesselY (ν, z)$

Floating-point Evaluation

>	$HankelH1 (0.3, 0.5), BesselJ (0.3, 0.5) + I BesselY (0.3, 0.5)$

$0.7002604885 - 0.8080475075 I, 0.7002604885 - 0.8080475075 I$

(3.1.1)

Differentiation

The derivatives of the Bessel and Hankel functions may be expressed as algebraic expressions in themselves with shifted parameters:

>	$\frac{\partial}{\partial z} BesselJ (ν, z)$

$- J_{ν + 1} (z) + \frac{ν J_{ν} (z)}{z}$

(3.2.1)

>	$\frac{&DifferentialD;}{&DifferentialD; z} BesselY (1, z)$

$Y_{0} (z) - \frac{Y_{1} (z)}{z}$

(3.2.2)

>	$\frac{\partial}{\partial z} BesselI (ν, z)$

$I_{ν + 1} (z) + \frac{ν I_{ν} (z)}{z}$

(3.2.3)

>	$\frac{\partial}{\partial z} BesselK (ν, z)$

$- K_{ν + 1} (z) + \frac{ν K_{ν} (z)}{z}$

(3.2.4)

>	$\frac{\partial}{\partial z} HankelH1 (ν, z)$

$- H_{ν + 1}^{(1)} (z) + \frac{ν H_{ν}^{(1)} (z)}{z}$

(3.2.5)

>	$\frac{\partial}{\partial z} HankelH2 (ν, z)$

$- H_{ν + 1}^{(2)} (z) + \frac{ν H_{ν}^{(2)} (z)}{z}$

(3.2.6)

Graphing

>	$plot ([BesselJ (10.2, z), BesselY (10.2, z), \sqrt{{BesselJ (10.2, z)}^{2} + {BesselY (10.2, z)}^{2}}], z = 0 .. 30, legend = [J, Y, sqrt(J^2+Y^2)], view = [0 .. 30, - 0.6 .. 0.6])$

>	$plot ([seq (BesselJ (i, z), i = 0 .. 6)], z = 0 .. 10)$

>	$plot ([seq (BesselJ (ν, i), i = 0 .. 6)], ν = - 4 .. 10, view = [- 4 .. 10, - 1 .. 1], numpoints = 100)$

The Kummer (Confluent Hypergeometric) Functions

The Kummer M and U functions, KummerM(mu, nu, z) and KummerU(mu, nu, z), are solutions of the differential equation
$z (\frac{\partial^{2}}{\partial z \partial z} y (z)) + (ν - z) y (z) - μ y (z) = 0$
which can be written as:

$KummerM (μ, ν, z) = \sum_{k = 0}^{\infty} \frac{pochhammer (μ, k) z^{k}}{pochhammer (ν, k) k &excl;}$

Floating-point Evaluation

Asymptotic approximations:

>	$KummerM (0.3, 0.5, 0.1), \frac{Γ (0.5) {&ExponentialE;}^{0.1} {0.1}^{0.3 - 0.5}}{Γ (0.3)}$

$1.062681649, 1.037780156$

(4.1.1)

>	$KummerM (0.3, 0.5, 100.0), \frac{Γ (0.5) {&ExponentialE;}^{100.0} {100.0}^{0.3 - 0.5}}{Γ (0.3)}$

$6.349477824 10^{42}, 6.340508722 10^{42}$

(4.1.2)

>	$KummerM (0.3, 0.5, - 0.1), \frac{Γ (0.5) {0.1}^{- 0.3}}{Γ (0.5 - 0.3)}$

$0.9425221107, 0.7703399627$

(4.1.3)

>	$KummerM (0.3, 0.5, - 100.0), \frac{Γ (0.5) {100.0}^{- 0.3}}{Γ (0.5 - 0.3)}$

$0.09721559118, 0.09698005550$

(4.1.4)

Differentiation

The derivatives of the Kummer functions may be expressed as algebraic expressions in themselves with shifted parameters:

>	$\frac{\partial}{\partial z} KummerM (μ, ν, z)$

$\frac{(z + μ - ν) M (μ, ν, z) + (ν - μ) M (- 1 + μ, ν, z)}{z}$

(4.2.1)

>	$\frac{\partial}{\partial z} KummerU (μ, ν, z)$

$\frac{(z + μ - ν) U (μ, ν, z) - U (- 1 + μ, ν, z)}{z}$

(4.2.2)

Graphing

>	$plot ([seq (KummerM (\frac{i}{2}, 0.5, z), i = - 8 .. 0)], z = 0 .. 8, view = [0 .. 8, - 10 .. 14], color = [seq (COLOR (RGB, \frac{i}{8}, 0, 1 - \frac{i}{8}), i = 0 .. 8)])$

>	$plot3d (KummerM (μ, ν, 1), μ = - 5 .. 5, ν = - 5 .. 5, view = [- 5 .. 5, - 5 .. 5, - 5 .. 5], grid = [100, 100], style = PATCHNOGRID, axes = framed)$

The Parabolic Cylinder Functions

The cylinder U and V functions, CylinderU(a, z) and CylinderV(a,z) respectively, are solutions to the differential equation:
$\frac{\partial^{2}}{\partial z \partial z} y (z) - (\frac{z^{2}}{4} + a) y (z) = 0$

The Whittaker's parabolic D function, CylinderD(a,z), is related to the above by the following expression

$CylinderD (- a - \frac{1}{2}, z) = CylinderU (a, z)$
which are equal when $a = - \frac{1}{4}$ .

Floating-point Evaluation

>	$CylinderU (3.2, 1.2), CylinderD (- 3.7, 1.2), CylinderV (3.2, 1.2)$

$0.04426305425, 0.04426305425, 4.724868148$

(5.1.1)

>	$CylinderD (- 0.25, - 5.2), CylinderU (- 0.25, - 5.2)$

$177.8936549, 177.8936549$

(5.1.2)

Differentiation

The derivatives of the cylinder functions can be written as simple algebraic expressions of themselves with modified parameters:

>	$\frac{\partial}{\partial z} CylinderU (a, z)$

$- \frac{z U (a, z)}{2} - (a + \frac{1}{2}) U (a + 1, z)$

(5.2.1)

>	$\frac{\partial}{\partial z} CylinderV (a, z)$

$- \frac{z V (a, z)}{2} + V (a + 1, z)$

(5.2.2)

The derivative of the Whittaker parabolic D function is cleaner than the derivative of the cylinder U function:

>	$\frac{\partial}{\partial z} CylinderD (a, z)$

$\frac{z D_{a} (z)}{2} - D_{a + 1} (z)$

(5.2.3)

Graphing

>	$plot ([' CylinderU (1.3, z), CylinderV (1.3, z), CylinderD (1.3, z)'], z = - 1 .. 2, legend = [U, V, D])$

$CyU := plot3d (' CylinderU (a, z)', a = - 2 .. 2, z = - 1 .. 2, color = red) &colon;$ $CyV ≔ plot3d (' CylinderV (a, z)', a = - 2 .. 2, z = - 1 .. 2, color = blue) &colon;$ $CyD ≔ plot3d (' CylinderD (a, z)', a = - 2 .. 2, z = - 1 .. 2, color = green) &colon;$ $plots [display] (CyU, CyV, CyD, axes = framed)$

The Elliptic Functions

The incomplete elliptic F function of the first kind, EllipticF(z,k), and complete elliptic K and CK functions of the first kind, EllipticK(k) and EllipticCK(k), are defined as:
           $EllipticF (z, k) = \int_{0}^{z} \frac{1}{\sqrt{1 - t^{2}} \sqrt{1 - k^{2} t^{2}}} &DifferentialD; t$
           $E$ $llipticK (k) = EllipticF (1, k)$
      $EllipticCK (k) = EllipticF (1, \sqrt{1 - k^{2}})$
The incomplete elliptic E function of the second kind, EllipticE(z,k), and complete elliptic E and CE functions of the second kind, EllipticE(k) and EllipticCE(k), are defined as:
           $EllipticE (z, k) = \int_{0}^{z} \frac{\sqrt{1 - k^{2} t^{2}}}{\sqrt{1 - t^{2}}} &DifferentialD; t$
           $EllipticE (k) = EllipticE (1, k)$
      $EllipticCE (k) = EllipticE (1, \sqrt{1 - k^{2}})$
The incomplete elliptic Pi function of the third kind, EllipticPi(z,nu,k), and complete elliptic Pi and CPi functions of the second kind, EllipticPi(nu,k) and EllipticCPi(nu,k), are defined as:
           $EllipticPi (z, ν, k) = \int_{0}^{z} \frac{1}{(1 - ν t^{2}) \sqrt{1 - t^{2}} \sqrt{1 - k^{2} t^{2}}} &DifferentialD; t$
           $EllipticPi (ν, k) = EllipticPi (1, ν, k)$
      $EllipticCPi (ν, k) = EllipticPi (1, ν, \sqrt{1 - k^{2}})$

Floating-point Evaluation

>	$EllipticK (0.2), EllipticF (1.0, 0.2)$

$1.586867847, 1.586867847 + −0. I$

(6.1.1)

>	$EllipticF (0.8, 0.2), EllipticPi (0.8, 0.0, 0.2)$

$0.9318236075, 0.9318236075$

(6.1.2)

>	$EllipticPi (1.0, 0.7, 0.2), EllipticPi (0.7, 0.2)$

$2.905870485, 2.905870485$

(6.1.3)

Differentiation

The derivatives of the elliptic functions of the first and second kinds are in the form of algebraic expressions in themselves:

>	$\frac{&DifferentialD;}{&DifferentialD; z} EllipticK (z)$

$\frac{E (z)}{(- z^{2} + 1) z} - \frac{K (z)}{z}$

(6.2.1)

>	$\frac{&DifferentialD;}{&DifferentialD; z} EllipticE (z)$

$\frac{E (z)}{z} - \frac{K (z)}{z}$

(6.2.2)

The derivatives of the elliptic functions of the third kind are algebraic expressions in all three kinds of elliptic functions:

>	$\frac{&DifferentialD;}{&DifferentialD; z} EllipticPi (z, 0)$

$\frac{π}{4 {(1 - z)}^{\frac{3}{2}}}$

(6.2.3)

>	$\frac{\partial}{\partial z} EllipticPi (z, ν)$

$\frac{Π (z, ν) (ν^{2} - z^{2})}{2 (1 - z) (ν^{2} - z) z} - \frac{K (ν)}{2 (1 - z) z} - \frac{E (ν)}{2 (1 - z) (ν^{2} - z)}$

(6.2.4)

Graphing

>	$plot ([EllipticK (z), EllipticE (z), EllipticPi (z, 0.5)], z = 0 .. 1, legend = [K, E, Pi], view = [0 .. 1, 1 .. 3])$

>	$plot3d (\{EllipticF (z, k), EllipticE (z, k), EllipticPi (z, 0.5, k)\}, z = 0 .. 0.8, k = 0.8 .. 1, axes = framed)$

The Jacobi Elliptic Functions

The Jacobi sn, cn, dn, ..., dc functions, JacobiSN(z,k), JacobiCN(z,k), JacobiDN(z,k), ..., JacobiDC(z,k) are defined in terms of the Jacobi am amplitude function JacobiAM(z, k):
           $JacobiSN (z, k) = \sin (JacobiAM (z, k))$
           $JacobiCN (z, k) = \cos (JacobiAM (z, k))$
           $JacobiDN (z, k) = \sqrt{1 - k^{2} {JacobiSN (z, k)}^{2}}$
                                    .
                                    .
                                    .
           $JacobiDC (z, k) = \frac{JacobiDN (z, k)}{JacobiCN (z, k)}$

Floating-point Evaluation

>	$JacobiSN (0.3, 0.), JacobiCN (0.3, 0.), JacobiDN (0.3, 0.)$

$0.2955202067, 0.9553364891, 1.$

(7.1.1)

>	$\sin (0.3), \cos (0.3), 1.$

$0.2955202067, 0.9553364891, 1.$

(7.1.2)

>	$JacobiSN (0.3, 1.), JacobiCN (0.3, 1.), JacobiDN (0.3, 1.)$

$0.2913126125, 0.9566279119, 0.9566279119$

(7.1.3)

>	$\tanh (0.3), sech (0.3)$

$0.2913126125, 0.9566279119$

(7.1.4)

Differentiation

The derivatives of the Jacobi elliptic functions are expressions in other Jacobi elliptic functions:

>	$\frac{\partial}{\partial z} JacobiSN (z, k)$

$cn (z | k) dn (z | k)$

(7.2.1)

>	$\frac{\partial}{\partial z} JacobiDC (z, k)$

$\frac{sc (z | k) (- k^{2} + 1)}{cn (z | k)}$

(7.2.2)

Graphing

>	$plot ([JacobiSN (z, 0.25), JacobiCN (z, 0.25), JacobiDN (z, 0.25)], z = 0 .. 4 EllipticK (0.25), legend = [sn, cn, dn])$

>	$plot ([JacobiSC (z, 0.5), JacobiCS (z, 0.5), JacobiCD (z, 0.5), JacobiDC (z, 0.5)], z = 0 .. 4 EllipticK (0.5), legend = [sc, cs, cd, dc], view = [0 .. 4 EllipticK (0.5), - 2 .. 2], discont = true)$

The Jacobi Theta Functions

The Jacobi $θ_{1}$ , $θ_{2}$ , $θ_{3}$ , and $θ_{4}$ functions JacobiTheta1(z, q), JacobiTheta2(z, q), JacobiTheta3(z, q), and JacobiTheta4(z, q) are defined as:
$JacobiTheta1 (z, q) = 2 q^{\frac{1}{4}} (\sum_{n = 0}^{\infty} {(- 1)}^{n} q^{n (n + 1)} \sin ((2 n + 1) z))$

          $JacobiTheta2 (z, q) = 2 q^{\frac{1}{4}} (\sum_{n = 0}^{\infty} q^{n (n + 1)} \cos ((2 n + 1) z))$
           $JacobiTheta3 (z, q) = 1 + 2 (\sum_{n = 1}^{\infty} q^{n^{2}} \cos (2 n z))$
           $JacobiTheta4 (z, q) = 1 + 2 (\sum_{n = 1}^{\infty} {(- 1)}^{n} q^{n^{2}} \cos (2 n z))$

Floating-point Evaluation

>	${JacobiTheta2 (0, 0.2)}^{4} + {JacobiTheta4 (0, 0.2)}^{4}, {JacobiTheta3 (0, 0.2)}^{4}$

$3.876855127, 3.876855123$

(8.1.1)

>	$evalf (\binom{\frac{&DifferentialD;}{&DifferentialD; z} JacobiTheta1 (z, 0.4)}{} \| \binom{}{z = 0.0}), JacobiTheta2 (0, 0.4) JacobiTheta3 (0, 0.4) JacobiTheta4 (0, 0.4)$

$0.8594691945, 0.8594692103$

(8.1.2)

>	$JacobiTheta4 (0.0, 0.1), JacobiTheta4 (2 EllipticK (0.1), 0.1)$

$0.8001999980, 0.8002248520$

(8.1.3)

Differentiation

The derivatives of the Jacobi theta functions are less elegant than other special functions:

>	$\frac{\partial}{\partial z} JacobiTheta1 (z, q)$

$\frac{2 K (k (q)) (ϑ_{1} (z, q) Z (\frac{2 K (k (q)) (z - \frac{π}{2})}{π}, k (q)) + \frac{ϑ_{4} (z, q) ϑ_{2} (z, q)}{ϑ_{3} (z, q)})}{π}$

(8.2.1)

>	$\frac{\partial}{\partial z} JacobiTheta2 (z, q)$

$\frac{2 K (k (q)) (ϑ_{2} (z, q) Z (\frac{2 K (k (q)) z}{π}, k (q)) - \frac{ϑ_{3} (z, q) ϑ_{1} (z, q)}{ϑ_{4} (z, q)})}{π}$

(8.2.2)

Graphing

>	$plot ([JacobiTheta1 (z, 0.2), JacobiTheta2 (z, 0.2), JacobiTheta3 (z, 0.2), JacobiTheta4 (z, 0.2)], z = 0 .. 4 EllipticK (0.2), legend = [1, 2, 3, 4])$

>	$Re1 ≔ plot3d (Re (JacobiTheta1 (a + b I, 0.2)), a = 0 .. 4 EllipticK (0.2), b = - 1 .. 1, color = red) &colon;$ $Im1 ≔ plot3d (Im (JacobiTheta1 (a + b I, 0.2)), a = 0 .. 4 EllipticK (0.2), b = - 1 .. 1, color = blue) &colon;$ $plots [display] ([Re1, Im1], axes = framed)$

The Riemann and Hurwitz Zeta Functions

The Riemann $ζ$ function Zeta may be defined as
           $ζ (s) = \sum_{k = 1}^{\infty} \frac{1}{k^{s}}$
or
           $ζ (s) = \prod_{p} (\frac{1}{1 - \frac{1}{p^{s}}})$
where the product is over all primes p. The Hurwitz Zeta function is defined as:
           $ζ (0, s, ν) = \sum_{k = 0}^{\infty} \frac{1}{{(k + ν)}^{s}}$

Floating-point Evaluation

$ζ (1.3)$

$3.931949212$

(9.1.1)

>	$ζ (0, 1.3, 1)$

$3.931949212$

(9.1.2)

>	$ζ (0, 1.3, 1.1)$

$3.761371026$

(9.1.3)

Differentiation

The derivative of the Zeta function is denoted as Zeta(1,z):

>	$\frac{&DifferentialD;}{&DifferentialD; z} ζ (z)$

$ζ^{(1)} (z)$

(9.2.1)

>	$\frac{\partial}{\partial z} ζ (0, z, ν)$

$ζ^{(1)} (z, ν)$

(9.2.2)

Graphing

>	$plot (ζ (z), z = - 10 .. 10, view = [- 10 .. 10, - 10 .. 10], discont = true)$

>	$plot ([Re (ζ (\frac{1}{2} + I z)), Im (ζ (\frac{1}{2} + I z))], z = - 20 .. 20, legend = [Real Part, Imaginary Part])$

The Jacobi Zeta Function

The Jacobi Zeta function is defined by
$JacobiZeta (z, k) = \frac{\partial}{\partial z} \ln (JacobiTheta4 (\frac{π z}{2 EllipticK (k)}, EllipticNome (k)))$

Floating-point Evaluation

>	$JacobiZeta (0.2, 0.3)$

$0.008868168440$

(10.1.1)

Differentiation

The derivatives of the Jacobi Zeta function can be written in terms of elliptic functions:

>	$\frac{\partial}{\partial z} JacobiZeta (z, k)$

${dn (z | k)}^{2} - \frac{E (k)}{K (k)}$

(10.2.1)

Graphing

>	$plot ([seq (JacobiZeta (z, \frac{i}{10}), i = 1 .. 9)], z = 0 .. 4 EllipticK (0.))$

>	$plot3d (JacobiZeta (z, k), z = 0 .. 10, k = - 0.99 .. 0.99,' axes = framed',' grid' = [100, 100],' style = PATCHNOGRID',' lightmodel = light4')$

The Kelvin Functions

The Kelvin Ber and Bei functions, KelvinBer(nu, z) and KelvinBei(nu,z), are defined by the following equations:
$KelvinBer (ν, z) + I KelvinBei (ν, x) = BesselJ (ν, z {&ExponentialE;}^{\frac{3 I π}{4}})$

$KelvinBer (ν, z) - I KelvinBei (ν, x) = BesselJ (ν, z {&ExponentialE;}^{- \frac{3 I π}{4}})$

The Kelvin Ker and Kei functions, KelvinKer(nu,z) and KelvinKei(nu,z), are defined by the following equations:
$KelvinKer (ν, z) + I KelvinKei (ν, x) = {&ExponentialE;}^{- \frac{ν π I}{2}} BesselK (ν, z {&ExponentialE;}^{\frac{I π}{4}})$

$KelvinKer (ν, z) - I KelvinKei (ν, z) = {&ExponentialE;}^{- \frac{ν π I}{2}} BesselK (ν, z {&ExponentialE;}^{- \frac{I π}{4}})$

The Kelvin Her and Hei functions, KelvinHer(nu,z) and KelvinHei(nu,z), are defined by the following equations:
$KelvinHer (ν, z) + I KelvinHei (ν, z) = HankelH1 (ν, z {&ExponentialE;}^{\frac{3 I π}{4}})$

$KelvinHer (ν, z) - I KelvinHei (ν, z) = HankelH2 (ν, z {&ExponentialE;}^{- \frac{3 I π}{4}})$

Floating-point Evaluation

>	$KelvinBer (1.0, 3.2) + I KelvinBei (1.0, 3.2), BesselJ (1.0, 3.2 {&ExponentialE;}^{\frac{3 I π}{4}})$

$−1.848052313 - 0.7875000586 I, −1.848052312 - 0.7875000568 I$

(11.1.1)

>	$WeberE (1.0, 0.5), evalf (\frac{2}{π}) - StruveH (1.0, 0.5)$

$0.5844460281, 0.5844460280$

(11.1.2)

Differentiation

The derivatives of the Kelvin functions can be written in terms of algebraic expressions in themselves with shifted parameters:

>	$\frac{\partial}{\partial z} KelvinBer (ν, z)$

$\frac{\sqrt{2} ({ber}_{ν + 1} (z) + {bei}_{ν + 1} (z))}{2} + \frac{ν {ber}_{ν} (z)}{z}$

(11.2.1)

>	$\frac{\partial}{\partial z} KelvinKer (ν, z)$

$\frac{\sqrt{2} (\ker_{ν + 1} (z) + {kei}_{ν + 1} (z))}{2} + \frac{ν \ker_{ν} (z)}{z}$

(11.2.2)

>	$\frac{\partial}{\partial z} KelvinHer (ν, z)$

$\frac{\sqrt{2} ({her}_{ν + 1} (z) + {hei}_{ν + 1} (z))}{2} + \frac{ν {her}_{ν} (z)}{z}$

(11.2.3)

Graphing

>	$plot ([KelvinBer (2.5, z), KelvinBei (2.5, z), KelvinKer (2.5, z), KelvinKei (2.5, z), KelvinHer (2.5, z), KelvinHei (2.5, z)], z = 0 .. 10, legend = [Ber, Bei, Ker, Kei, Her, Hei], view = [0 .. 10, - 10 .. 10])$

>	$plot ([seq (KelvinBer (\frac{i}{2}, z), i = 0 .. 4)], z = 0 .. 10, legend = [Ber 0, Ber 1/2, Ber 1, Ber 3/2, Ber 2], view = [0 .. 10, - 50 .. 50])$

>	$plot3d (KelvinBer (ν, z), ν = 0 .. 10, z = 0 .. 5, axes = framed)$

The Legendre Functions

The Legendre P and Q functions of the first and second kinds, LegendreP(nu, z) and LegendreQ(nu, z), solve the differential equation:
${(1 - z)}^{2} (\frac{\partial^{2}}{\partial z \partial z} y (z)) - 2 z (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) + ν (ν + 1) y (z) = 0$

The associated Legendre P and Q functions of the first and second kinds, LegendreP(nu, mu, z) and LegendreQ(nu, mu, z), solve the differential equation:

${(1 - z)}^{2} (\frac{\partial^{2}}{\partial z \partial z} y (z)) - 2 z (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) + (ν (ν + 1) - \frac{μ^{2}}{1 - z^{2}}) y (z) = 0$
The associated case simplifies to the regular case when $μ = 0$ .

Floating-point Evaluation

>	$LegendreP (3.2, 5.2), LegendreP (3.2, 0.0, 5.2)$

$531.9471184, 531.9471184$

(12.1.1)

>	$LegendreQ (3.2, 0.1, 5.2), LegendreQ (3.2, 5.2)$

$0.00005407404071 + 0.00001756972088 I, 0.00004980176423$

(12.1.2)

Differentiation

The derivative of the Legendre functions can be written as algebraic expressions in themselves with shifted parameters:

>	$\frac{\partial}{\partial z} LegendreP (ν, z)$

$\frac{(ν + 1) P_{ν + 1} (z) - (ν + 1) z P_{ν} (z)}{z^{2} - 1}$

(12.2.1)

>	$\frac{\partial}{\partial z} LegendreQ (ν, z)$

$\frac{(ν + 1) Q_{ν + 1} (z) - (ν + 1) z Q_{ν} (z)}{z^{2} - 1}$

(12.2.2)

>	$\frac{\partial}{\partial z} LegendreP (ν, μ, z)$

$\frac{(ν - μ + 1) P_{ν + 1}^{μ} (z) - (ν + 1) z P_{ν}^{μ} (z)}{z^{2} - 1}$

(12.2.3)

>	$\frac{\partial}{\partial z} LegendreQ (ν, μ, z)$

$\frac{(ν - μ + 1) Q_{ν + 1}^{μ} (z) - (ν + 1) z Q_{ν}^{μ} (z)}{z^{2} - 1}$

(12.2.4)

Graphing

>	$plot ([LegendreP (1.5, z), LegendreQ (1.5, z)], z = - 3 .. 3, legend = [P, Q])$

>	$plot3d (\{LegendreP (ν, z), LegendreQ (ν, z)\}, ν = - 2 .. 2, z = 1 .. 3, axes = framed, view = [- 2 .. 2, 1 .. 3, - 5 .. 5])$

The Lommel Functions

The Lommel s and S functions, LommelS1(mu, nu, z) and LommelS2(nu, z), are defined by the following equations:
$z^{2} (\frac{\partial^{2}}{\partial z \partial z} y (z)) + z (\frac{&DifferentialD;}{&DifferentialD; z} y (z)) + (z^{2} - ν^{2}) y (z) = z^{μ + 1}$

Floating-point Evaluation

The difference between the Anger J and Bessel J functions is approximately equal to the second expression in terms of Lommel S functions:

>	$L1 ≔ AngerJ (ν, z) - BesselJ (ν, z), \frac{\sin (ν π) LommelS1 (0, ν, z)}{π z} - \frac{ν \sin (ν π) LommelS1 (- 1, ν, z)}{π z^{2}}$

$L1 ≔ J_{ν} (z) - J_{ν} (z), \frac{\sin (ν π) (\tan (\frac{ν π}{2}) J_{ν} (z) - E_{ν} (z))}{2 z} + \frac{\sin (ν π) (\cot (\frac{ν π}{2}) J_{ν} (z) + E_{ν} (z))}{2 z^{2}}$

(13.1.1)

>	$evalf (eval (L1, [ν = 1.5, z = 100.0]))$

$−0.00313575834, −0.0005779109147$

(13.1.2)

>	$L2 ≔ WeberE (ν, z) + BesselY (ν, z), - \frac{(1 + \cos (ν π)) LommelS2 (0, ν, z)}{π ν} - \frac{ν (1 - \cos (ν π)) LommelS2 (- 1, ν, z)}{π z^{2}}$

$L2 ≔ E_{ν} (z) + Y_{ν} (z), - \frac{(1 + \cos (ν π)) ((J_{ν} (z) - J_{ν} (z)) \tan (\frac{ν π}{2}) - Y_{ν} (z) - E_{ν} (z))}{2 ν} - \frac{(1 - \cos (ν π)) ((J_{ν} (z) - J_{ν} (z)) \cot (\frac{ν π}{2}) - Y_{ν} (z) - E_{ν} (z))}{2 z^{2}}$

(13.1.3)

>	$evalf (eval (L2, [ν = 1.5, z = 100.2]))$

$−0.00322468841, −0.002118098484$

(13.1.4)

Differentiation

The derivative of the Lommel functions can be written in terms of algebraic expressions in themselves with shifted parameters:

>	$\frac{\partial}{\partial z} LommelS1 (μ, ν, z)$

$- \frac{ν s_{μ, ν}^{(+)} (z)}{z} + (μ + ν - 1) s_{μ - 1, - 1 + ν}^{(+)} (z)$

(13.2.1)

>	$\frac{\partial}{\partial z} LommelS2 (μ, ν, z)$

$- \frac{ν s_{μ, ν}^{(-)} (z)}{z} + (μ + ν - 1) s_{μ - 1, - 1 + ν}^{(-)} (z)$

(13.2.2)

Graphing

>	$plot3d (LommelS2 (1.5, ν, z), ν = 0 .. 5, z = 0 .. 5, axes = framed, grid = [100, 100], view = [0 .. 5, 0 .. 5, - 10 .. 100], style = PATCHNOGRID)$

The Weierstrass Functions

The Weierstrass $P$ , $P'$ , $ζ$ , and $σ$ functions, WeierstrassP(z, g2, g3), WeierstrassPPrime(z, g2, g3), WeierstrassZeta(z, g2, g3), and WeierstrassSigma(z, g2, g3) may be defined as follows:
           $WeierstrassP (z, g2, g3) = \frac{1}{z^{2}} + \sum_{w} (\frac{1}{{(z - w)}^{2}} - \frac{1}{w^{2}})$
           $WeierstrassPPrime (z, g2, g3) = \frac{\partial}{\partial z} WeierstrassPPrime (z, g2, g3)$
            $WeierstrassZeta (z, g2, g3) = - (\int_{0}^{z} WeierstrassP (t, g2, g3) &DifferentialD; t)$

and,
           $WeiertrassSigma (z, g2, g3) = {&ExponentialE;}^{\int_{0}^{z} WeierstrassZeta (t, g2, g3) &DifferentialD; t}$

Floating-point Evaluation

>	$WeierstrassP (0.2, 0.8, 0.6), \frac{WeierstrassP (0.1, 0.4, 0.3)}{{2.}^{2}}$

$25.00163432, 25.0000502678311$

(14.1.1)

>	$WeierstrassPPrime (0.2, 0.4, 0.6), \frac{WeierstrassPPrime (0.1, 0.2, 0.3)}{{2.}^{3}}$

$−249.9913140, −249.9997446$

(14.1.2)

>	$WeierstrassZeta (0.2, 0.8, 0.6), \frac{WeierstrassZeta (0.1, 0.4, 0.3)}{2.}$

$4.999891962, 4.99999665666235$

(14.1.3)

>	$WeierstrassSigma (0.2, 0.4, 0.6), WeierstrassSigma (0.1, 0.2, 0.3) \cdot 2.$

$0.1999994576, 0.1999999833$

(14.1.4)

Differentiation

The derivatives of the various Weierstrass functions can be written in terms of simple algebraic expressions of other Weierstrass functions:

>	$\frac{\partial}{\partial z} WeierstrassP (z, g2, g3)$

$𝒫' (z; g2, g3)$

(14.2.1)

>	$\frac{\partial}{\partial z} WeierstrassPPrime (z, g2, g3)$

$6 {𝒫 (z; g2, g3)}^{2} - \frac{g2}{2}$

(14.2.2)

>	$\frac{\partial}{\partial z} WeierstrassZeta (z, g2, g3)$

$- 𝒫 (z; g2, g3)$

(14.2.3)

>	$\frac{\partial}{\partial z} WeierstrassSigma (z, g2, g3)$

$ζ (z; g2, g3) σ (z; g2, g3)$

(14.2.4)

Graphing

>	$plot ([WeierstrassP (z, 0.5, 0.1), WeierstrassPPrime (z, 0.5, 0.1), WeierstrassZeta (z, 0.5, 0.1), WeierstrassSigma (z, 0.5, 0.1)], z = - 3 .. 3, legend = [P, P', zeta, sigma], view = [- 3 .. 3, - 10 .. 10])$

>	$plot3d (Re (WeierstrassP (a + b I, 0.3, 0.5)), a = - 2 .. 2, b = - 2 .. 2, axes = framed, view = [- 2 .. 2, - 2 .. 2, - 4 .. 4])$

The Whittaker Functions

The Whittaker M and W functions, WhittakerM(mu, nu, z) and WhittakerW(mu, nu, z), solve the differential equation:
$\frac{\partial^{2}}{\partial z \partial z} y (z) + (- \frac{1}{4} + \frac{μ}{z} + \frac{\frac{1}{4} - ν^{2}}{z^{2}}) y (z) = 0$

They are related to each other by:

$WhittakerW (μ, ν, z) = \frac{Γ (- 2 ν) WhittakerM (μ, ν, z)}{Γ (\frac{1}{2} - ν - μ)} + \frac{Γ (2 ν) WhittakerM (μ, - ν, z)}{Γ (\frac{1}{2} + ν - μ)}$

Floating-point Evaluation

>	$WhittakerW (0.3, 0.1, 1.2)$

$0.5694807360$

(15.1.1)

>	$\frac{Γ (- 2 \cdot 0.1) WhittakerM (0.3, 0.1, 1.2)}{Γ (\frac{1}{2} - 0.3 - 0.1)} + \frac{Γ (2 \cdot 0.1) WhittakerM (0.3, - 0.1, 1.2)}{Γ (\frac{1}{2} + 0.1 - 0.3)}$

$0.5694807362$

(15.1.2)

Differentiation

The derivatives of the Whittaker functions can be written as algebraic expressions in themselves with a shifted first parameter:

>	$\frac{\partial}{\partial z} WhittakerM (μ, ν, z)$

$(\frac{1}{2} - \frac{μ}{z}) M_{μ, ν} (z) + \frac{(\frac{1}{2} + ν + μ) M_{μ + 1, ν} (z)}{z}$

(15.2.1)

>	$\frac{\partial}{\partial z} WhittakerW (μ, ν, z)$

$(\frac{1}{2} - \frac{μ}{z}) W_{μ, ν} (z) - \frac{W_{μ + 1, ν} (z)}{z}$

(15.2.2)

Graphing

>	$plot ([WhittakerM (1.3, 0.5, z), WhittakerW (1.3, 0.5, z)], z = 0 .. 10, legend = [M, W])$

>	$plot ([seq (seq (WhittakerW (1.3 + i \cdot 0.2, 1.5 + j \cdot 0.2, z), i = 0 .. 1), j = 0 .. 1)], z = 0 .. 10, view = [0 .. 10, 0 .. 6])$

>	$WhM := plot3d (WhittakerM (μ, ν, 1.0), μ = 0 .. 1, ν = 0 .. 1, color = red) &colon;$ $WhW ≔ plot3d (WhittakerW (μ, ν, 1.0), μ = 0 .. 1, ν = 0 .. 1, color = blue) &colon;$ $plots [display] ([WhM, WhW], axes = framed)$

For more information, consult the help pages for the individual functions.

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