Weierstrass%0Aelliptic functions

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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: 11.9 Lommel Functions

§11.9 Lommel Functions

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Reflection Formulas

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§11.9(ii) Expansions in Series of Bessel Functions

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13: 20.2 Definitions and Periodic Properties

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§20.2(i) Fourier Series

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§20.2(ii) Periodicity and Quasi-Periodicity

… ►For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. ►The four points $(0,\pi ,\pi +\tau \pi ,\tau \pi )$ are the vertices of the fundamental parallelogram in the $z$-plane; see Figure 20.2.1. … ►

§20.2(iv) $z$-Zeros

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14: 31.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►

$x$, $y$	real variables.
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►The main functions treated in this chapter are $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$, $(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$, $(s_1,s_2){\mathrm{𝐻𝑓}}_m^{\nu }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$, and the polynomial ${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$. …Sometimes the parameters are suppressed.

15: 4.2 Definitions

… ►This is a multivalued function of $z$ with branch point at $z=0$. … ► $\ln z$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },0]$ and real-valued when $z$ ranges over the positive real numbers. … ►

§4.2(iii) The Exponential Function

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§4.2(iv) Powers

… ►In all other cases, $z^a$ is a multivalued function with branch point at $z=0$. …

16: 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).

17: 25.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►►

$k,m,n$	nonnegative integers.
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primes	on function symbols: derivatives with respect to argument.

►The main function treated in this chapter is the Riemann zeta function $\zeta \left(s\right)$. … ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

18: 1.10 Functions of a Complex Variable

… ►An analytic function $f(z)$ has a zero of order (or multiplicity) $m$ ($\ge 1$) at $z_0$ if the first nonzero coefficient in its Taylor series at $z_0$ is that of ${(z-z_0)}^m$. … ►Lastly, if $a_n\ne 0$ for infinitely many negative $n$, then $z_0$ is an isolated essential singularity. … ►

$M$-test

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Weierstrass Product

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§1.10(xi) Generating Functions

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19: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

… ►As a function of $a$, with $s$ ($\ne 1$) fixed, $\zeta (s,a)$ is analytic in the half-plane $\mathrm{\Re }a>0$. … ►Most references treat real $a$ with . … ►Throughout this subsection $\mathrm{\Re }a>0$. … ►As $a\to 0$ with $s$ $(\ne 1)$ fixed, …

20: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►(11.10.4) also applies when $\mathrm{\Re }z=0$ and $\mathrm{\Re }\nu >0$. … ►

§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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