mean value property for harmonic functions

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11: 14.20 Conical (or Mehler) Functions

… ► ►Another real-valued solution

\widehat{\mathsf{Q}}^{-\mu}_{-\frac{1}{2}+\mathrm{i}\tau}\left(x\right)

of (14.20.1) was introduced in Dunster (1991). … ►Lastly, for the range

1<x<\infty

P^{-\mu}_{-\frac{1}{2}+i\tau}\left(x\right)

is a real-valued solution of (14.20.1); in terms of

Q^{\mu}_{-\frac{1}{2}\pm\mathrm{i}\tau}\left(x\right)

(which are complex-valued in general): … ►where the inverse trigonometric functions take their principal values. … ►with the inverse tangent taking its principal value. …

12: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

►

§14.19(i) Introduction

… ►Most required properties of toroidal functions come directly from the results for

P^{\mu}_{\nu}\left(x\right)

and

\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)

. … ►

§14.19(iv) Sums

… ►

§14.19(v) Whipple’s Formula for Toroidal Functions

…

13: 9.1 Special Notation

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

$k$	nonnegative integer, except in §9.9(iii).
…

►The main functions treated in this chapter are the Airy functions

\operatorname{Ai}\left(z\right)

and

\operatorname{Bi}\left(z\right)

, and the Scorer functions

\operatorname{Gi}(z)

and

\operatorname{Hi}(z)

(also known as inhomogeneous Airy functions). ►Other notations that have been used are as follows:

\operatorname{Ai}\left(-x\right)

and

\operatorname{Bi}\left(-x\right)

for

\operatorname{Ai}\left(x\right)

and

\operatorname{Bi}\left(x\right)

(Jeffreys (1928), later changed to

\operatorname{Ai}\left(x\right)

and

\operatorname{Bi}\left(x\right)

);

U(x)=\sqrt{\pi}\operatorname{Bi}\left(x\right)

V(x)=\sqrt{\pi}\operatorname{Ai}\left(x\right)

(Fock (1945));

A(x)=3^{-\ifrac{1}{3}}\pi\operatorname{Ai}\left(-3^{-\ifrac{1}{3}}x\right)

(Szegő (1967, §1.81));

e_{0}(x)=\pi\operatorname{Hi}(-x)

\widetilde{e}_{0}(x)=-\pi\operatorname{Gi}(-x)

(Tumarkin (1959)).

14: 31.1 Special Notation

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

$x$ , $y$	real variables.
…

►The main functions treated in this chapter are

\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)

(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)

(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)

, and the polynomial

\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)

. …Sometimes the parameters are suppressed.

15: 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}}\left(\alpha;\beta,\beta^{\prime};\gamma;x,y\right)=\sum_{m,n=0}^{% \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},

\max\left(|x|,|y|\right)<1

ⓘ

Defines:: ${F_{1}}\left(\NVar{\alpha};\NVar{\beta},\NVar{\beta^{\prime}};\NVar{\gamma};% \NVar{x},\NVar{y}\right)$ : first Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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

… ►

16.13.4

{F_{4}}\left(\alpha,\beta;\gamma,\gamma^{\prime};x,y\right)=\sum_{m,n=0}^{% \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},

\sqrt{|x|}+\sqrt{|y|}<1

ⓘ

Defines:: ${F_{4}}\left(\NVar{\alpha},\NVar{\beta};\NVar{\gamma},\NVar{\gamma^{\prime}};% \NVar{x},\NVar{y}\right)$ : fourth Appell function
Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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

… ► …

16: 4.2 Definitions

… ►This is a multivalued function of

z

with branch point at

z=0

. … ►As a consequence, it has the advantage of extending regions of validity of properties of principal values. … ►

§4.2(iii) The Exponential Function

… ►

§4.2(iv) Powers

… ►The principal value is …

17: 1.10 Functions of a Complex Variable

… ►

Harmonic Functions

… ►

§1.10(vi) Multivalued Functions

►Functions which have more than one value at a given point

z

are called multivalued (or many-valued) functions. … ►The last condition means that given

\epsilon

(

>0

) there exists a number

a_{0}\in[a,b)

that is independent of

z

and is such that … ►

§1.10(xi) Generating Functions

…

18: 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).

19: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

… ►These expansions converge absolutely for all finite values of

z

. … ►

§11.10(vi) Relations to Other Functions

… ►

§11.10(vii) Special Values

… ►where the prime on the second summation symbols means that the first term is to be halved. …

20: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: … ►

§25.11(v) Special Values

… ►where

H_{n}

are the harmonic numbers: … ►uniformly with respect to bounded nonnegative values of

\alpha

. …