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11: 16.13 Appell Functions

§16.13 Appell Functions

►The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): ►

16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

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Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §16.13 and Ch.16

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12: 5.12 Beta Function

§5.12 Beta Function

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Euler’s Beta Integral

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Figure 5.12.1: $t$-plane. Contour for first loop integral for the beta function. Magnify

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Figure 5.12.2: $t$-plane. Contour for second loop integral for the beta function. Magnify

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Pochhammer’s Integral

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13: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

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§14.20(i) Definitions and Wronskians

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§14.20(ii) Graphics

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§14.20(x) Zeros and Integrals

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14: 4.2 Definitions

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§4.2(iii) The Exponential Function

… ►The function $\exp$ is an entire function of $z$, with no real or complex zeros. … ►

§4.2(iv) Powers

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15: 10.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$, it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

16: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

… ►When $p\le q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function. … ► … ►

§16.2(v) Behavior with Respect to Parameters

… ►When $p\le q+1$ and $z$ is fixed and not a branch point, any branch of ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ is an entire function of each of the parameters $a_1,\mathrm{\dots },a_p,b_1,\mathrm{\dots },b_q$.

17: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

… ►The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. … …

18: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►Each is an entire function of $z$ and $\nu$. … ►

§11.10(vi) Relations to Other Functions

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§11.10(viii) Expansions in Series of Products of Bessel Functions

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19: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

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Bessel Functions

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Confluent Hypergeometric Functions

… ►In this case there are no real turning points, and the solutions of (12.2.3), with $z$ replaced by $x$, oscillate on the entire real $x$-axis. … ►

§12.14(x) Modulus and Phase Functions

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20: 1.10 Functions of a Complex Variable

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Analytic Functions

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§1.10(vi) Multivalued Functions

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§1.10(vii) Inverse Functions

… ►is an entire function with zeros at $z_n$. … ►

§1.10(xi) Generating Functions

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