Whittaker equation

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1: 28.32 Mathematical Applications

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§28.32(ii) Paraboloidal Coordinates

… ►is separated in this system, each of the separated equations can be reduced to the Whittaker–Hill equation (28.31.1), in which $A,B$ are separation constants. …

2: 13.14 Definitions and Basic Properties

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§13.14(i) Differential Equation

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Whittaker’s Equation

… ►Standard solutions are: … ►

13.14.14 $$M_{\kappa ,\mu }\left(z\right)=z^{\mu +\frac{1}{2}}\left(1+O\left(z\right)\right),$$ $2\mu \ne -1,-2,-3,\mathrm{\dots }$.

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Symbols:

$O\left(x\right)$: order not exceeding, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.14.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §13.14(iii), §13.14 and Ch.13

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§13.14(v) Numerically Satisfactory Solutions

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3: 13.28 Physical Applications

… ►and $V_{\kappa ,\mu }^{(j)}(z)$, $j=1,2$, denotes any pair of solutions of Whittaker’s equation (13.14.1). …

4: Bibliography U

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Theory of the Whittaker-Hill equation. Proc. Roy. Soc. Edinburgh Sect. A 69, pp. 28–44.

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External Links:

MathReview (R. K. Rathy), ZentralBlatt

Referenced by:

§28.31(i), §28.31(ii), §28.32(ii).

Encodings:

BibTeX

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5: 28.31 Equations of Whittaker–Hill and Ince

§28.31 Equations of Whittaker–Hill and Ince

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§28.31(i) Whittaker–Hill Equation

… ►and constant values of $A,B,k$, and $c$, is called the Equation of Whittaker–Hill. …

6: 28.34 Methods of Computation

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Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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7: 13.18 Relations to Other Functions

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13.18.1 $$M_{0,\frac{1}{2}}\left(2z\right)=2\sinh z,$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/13.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(i), §13.18 and Ch.13

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13.18.3 $$M_{\kappa ,-\kappa -\frac{1}{2}}\left(z\right)={\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }.$$

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Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(i), §13.18 and Ch.13

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13.18.7 $$W_{-\frac{1}{4},\pm \frac{1}{4}}\left(z^2\right)={\mathrm{e}}^{\frac{1}{2}{z}^2}\sqrt{\pi z}\mathrm{erfc}\left(z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable

Referenced by:

Erratum (V1.0.3) for Equation (13.18.7)

Permalink:

http://dlmf.nist.gov/13.18.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.3):

Originally the left-hand side was given correctly as $W_{-\frac{1}{4},-\frac{1}{4}}\left(z^2\right)$; the equation is true also for $W_{-\frac{1}{4},+\frac{1}{4}}\left(z^2\right)$.

See also:

Annotations for §13.18(ii), §13.18 and Ch.13

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13.18.10 $$W_{0,\frac{1}{3}}\left(\frac{4}{3}{z}^{\frac{3}{2}}\right)=2\sqrt{\pi }{z}^{\frac{1}{4}}\mathrm{Ai}\left(z\right).$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iii), §13.18 and Ch.13

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13.18.11 $$W_{-\frac{1}{2}a,\pm \frac{1}{4}}\left(\frac{1}{2}{z}^2\right)=2^{\frac{1}{2}a}\sqrt{z}U(a,z),$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.18.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.18(iv), §13.18 and Ch.13

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8: 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: 13.27 Mathematical Applications

… ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i).

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

ⓘ

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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