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1: 30.2 Differential Equations

§30.2 Differential Equations

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§30.2(i) Spheroidal Differential Equation

… ► … ►The Liouville normal form of equation (30.2.1) is … ►

§30.2(iii) Special Cases

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2: 15.10 Hypergeometric Differential Equation

§15.10 Hypergeometric Differential Equation

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§15.10(i) Fundamental Solutions

… ►This is the hypergeometric differential equation. … ► … ►The $\left(\genfrac{}{}{0.0pt}{}{6}{3}\right)=20$ connection formulas for the principal branches of Kummer’s solutions are: …

3: 31.2 Differential Equations

§31.2 Differential Equations

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§31.2(i) Heun’s Equation

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§31.2(ii) Normal Form of Heun’s Equation

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§31.2(v) Heun’s Equation Automorphisms

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Composite Transformations

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4: 29.2 Differential Equations

§29.2 Differential Equations

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§29.2(i) Lamé’s Equation

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§29.2(ii) Other Forms

… ►Equation (29.2.10) is a special case of Heun’s equation (31.2.1).

5: 32.2 Differential Equations

§32.2 Differential Equations

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§32.2(i) Introduction

►The six Painlevé equations ${\text{P}}_{\text{I}}$–${\text{P}}_{\text{VI}}$ are as follows: … ►

§32.2(ii) Renormalizations

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6: 13.18 Relations to Other Functions

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§13.18(i) Elementary Functions

… ►When $\kappa =0$ the Whittaker functions can be expressed as modified Bessel functions. … ►

§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)

7: 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(v) Numerically Satisfactory Solutions

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8: 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. … ►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).

9: 13.16 Integral Representations

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§13.16(i) Integrals Along the Real Line

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13.16.4 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left(z\right)=\frac{\sqrt{z}{\mathrm{e}}^{-\frac{1}{2}z}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{-\kappa -\frac{1}{2}}{I}_{2\mu }\left(2\sqrt{zt}\right)dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

Erratum (V1.0.5) for Equation (13.16.4)

Permalink:

http://dlmf.nist.gov/13.16.E4

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

Originally the condition for the validity of this equation was stated incorrectly as $\mathrm{\Re }(\kappa -\mu )-\frac{1}{2}>0$. The correct condition is .

See also:

Annotations for §13.16(i), §13.16 and Ch.13

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§13.16(ii) Contour Integrals

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§13.16(iii) Mellin–Barnes Integrals

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10: 28.2 Definitions and Basic Properties

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§28.2(i) Mathieu’s Equation

… ►This is the characteristic equation of Mathieu’s equation (28.2.1). … ►

§28.2(iv) Floquet Solutions

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Distribution

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