Whittaker functions

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1: 13.27 Mathematical Applications

§13.27 Mathematical Applications

… ►The other group elements correspond to integral operators whose kernels can be expressed in terms of Whittaker functions. This identification can be used to obtain various properties of the Whittaker functions, including recurrence relations and derivatives. … ►

2: 13.28 Physical Applications

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§13.28(ii) Coulomb Functions

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3: 13.22 Zeros

§13.22 Zeros

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4: 13.18 Relations to Other Functions

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§13.18(i) Elementary Functions

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§13.18(ii) Incomplete Gamma Functions

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§13.18(iv) Parabolic Cylinder Functions

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§13.18(v) Orthogonal Polynomials

… ►For representations of Coulomb functions in terms of Whittaker functions see (33.2.3), (33.2.7), (33.14.4) and (33.14.7)

5: 13.24 Series

§13.24 Series

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§13.24(i) Expansions in Series of Whittaker Functions

►For expansions of arbitrary functions in series of $M_{\kappa ,\mu }\left(z\right)$ functions see Schäfke (1961b). ►

§13.24(ii) Expansions in Series of Bessel Functions

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13.24.2 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=2^{2\mu }{z}^{\mu +\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{p}_s^{(\mu )}(z){\left(2\sqrt{\kappa z}\right)}^{-2\mu -s}{J}_{2\mu +s}\left(2\sqrt{\kappa z}\right),$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $s$: nonnegative integer, $z$: complex variable and $p_s$: coefficents

Permalink:

http://dlmf.nist.gov/13.24.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.24(ii), §13.24 and Ch.13

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6: 13.1 Special Notation

… ►The main functions treated in this chapter are the Kummer functions $M(a,b,z)$ and $U(a,b,z)$, Olver’s function $𝐌(a,b,z)$, and the Whittaker functions $M_{\kappa ,\mu }\left(z\right)$ and $W_{\kappa ,\mu }\left(z\right)$. … ►

7: 13.23 Integrals

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§13.23(i) Laplace and Mellin Transforms

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§13.23(ii) Fourier Transforms

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§13.23(iii) Hankel Transforms

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§13.23(iv) Integral Transforms in terms of Whittaker Functions

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8: 13.25 Products

§13.25 Products

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13.25.1 $$M_{\kappa ,\mu }\left(z\right)M_{\kappa ,-\mu -1}\left(z\right)+\frac{(\frac{1}{2}+\mu +\kappa )(\frac{1}{2}+\mu -\kappa )}{4\mu (1+\mu ){(1+2\mu )}^2}{M}_{\kappa ,\mu +1}\left(z\right)M_{\kappa ,-\mu }\left(z\right)=1.$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.25

Permalink:

http://dlmf.nist.gov/13.25.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.25 and Ch.13

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9: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

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13.19.1 $$M_{\kappa ,\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}x}{x}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{x}^{-s},$$ $\mu -\kappa \ne -\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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13.19.2 $$M_{\kappa ,\mu }\left(z\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{z}^{-s}+\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\mathrm{e}}^{-\frac{1}{2}z\pm (\frac{1}{2}+\mu -\kappa )\pi \mathrm{i}}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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13.19.3 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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10: 13.14 Definitions and Basic Properties

… ►Standard solutions are: … ►The series … ► … ►

§13.14(vi) Wronskians

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§13.14(vii) Connection Formulas

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