Kummer functions

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1: 13.6 Relations to Other Functions

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§13.6(i) Elementary Functions

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§13.6(ii) Incomplete Gamma Functions

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§13.6(iv) Parabolic Cylinder Functions

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§13.6(vi) Generalized Hypergeometric Functions

… ►For representations of Coulomb functions in terms of Kummer functions see (33.2.4), (33.2.8) and (33.14.5).

2: 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 $W_{\kappa ,\mu }\left(z\right)$. … ►

3: 13.10 Integrals

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§13.10(i) Indefinite Integrals

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§13.10(ii) Laplace Transforms

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§13.10(iii) Mellin Transforms

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§13.10(iv) Fourier Transforms

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4: 13.12 Products

§13.12 Products

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13.12.1 $$M(a,b,z)M(-a,-b,-z)+\frac{a(a-b)z^2}{b^2(1-b^2)}M(1+a,2+b,z)M(1-a,2-b,-z)=1.$$

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Symbols:

$M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function and $z$: complex variable

Referenced by:

§13.12, §13.25

Permalink:

http://dlmf.nist.gov/13.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.12 and Ch.13

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5: 13.2 Definitions and Basic Properties

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§13.2(vi) Wronskians

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Kummer’s Transformations

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6: 13.31 Approximations

§13.31 Approximations

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§13.31(i) Chebyshev-Series Expansions

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13.31.3 $$z^{a}U(a,1+a-b,z)=\underset{n\to \mathrm{\infty }}{lim}\frac{A_n(z)}{B_n(z)}.$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer, $z$: complex variable, $A_n(z)$: expansion and $B_n(z)$: expansion

Permalink:

http://dlmf.nist.gov/13.31.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.31(iii), §13.31 and Ch.13

7: 13.30 Tables

§13.30 Tables

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8: 13.13 Addition and Multiplication Theorems

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§13.13(i) Addition Theorems for $M(a,b,z)$

►The function $M(a,b,x+y)$ has the following expansions: … ►The function $U(a,b,x+y)$ has the following expansions: … ►

13.13.12 $${\mathrm{e}}^y{\left(\frac{x+y}{x}\right)}^{1-b}\sum _{n=0}^{\mathrm{\infty }}\frac{{(-y)}^n}{n!x^n}U(a-n,b-n,x),$$ .

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: nonnegative integer, $x$: real variable and $y$: real variable

Permalink:

http://dlmf.nist.gov/13.13.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.13(ii), §13.13 and Ch.13

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§13.13(iii) Multiplication Theorems for $M(a,b,z)$ and $U(a,b,z)$

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9: 12.20 Approximations

… ►Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions $U(a,b,x)$ and $M(a,b,x)$ (§13.2(i)) whose regions of validity include intervals with endpoints $x=\mathrm{\infty }$ and $x=0$, respectively. …

10: 13.4 Integral Representations

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§13.4(i) Integrals Along the Real Line

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13.4.6 $$U(a,b,z)=\frac{{(-1)}^n{z}^{1-b-n}}{\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{𝐌(b-a,b,t){\mathrm{e}}^{-t}{t}^{b+n-1}}{t+z}dt,$$ , $n=0,1,2,\mathrm{\dots }$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.10(vi)

Permalink:

http://dlmf.nist.gov/13.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(i), §13.4 and Ch.13

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§13.4(ii) Contour Integrals

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§13.4(iii) Mellin–Barnes Integrals

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