piecewise%20continuous%20functions

SearchAdvancedHelp

(0.003 seconds)

11—20 of 964 matching pages

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

ⓘ

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

… ►

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

ⓘ

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

… ► …

12: 15.2 Definitions and Analytical Properties

… ►

§15.2(i) Gauss Series

►The hypergeometric function $F(a,b;c;z)$ is defined by the Gauss series … … ►On the circle of convergence, $|z|=1$, the Gauss series: … ►

§15.2(ii) Analytic Properties

…

13: 14.20 Conical (or Mehler) Functions

§14.20 Conical (or Mehler) Functions

►

§14.20(i) Definitions and Wronskians

… ► … ►

§14.20(ii) Graphics

… ►

§14.20(x) Zeros and Integrals

…

14: 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).

15: 4.2 Definitions

… ► … ►

§4.2(iii) The Exponential Function

… ►

§4.2(iv) Powers

►

Powers with General Bases

… ► …

16: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

… ►

§8.17(ii) Hypergeometric Representations

… ►

§8.17(iii) Integral Representation

… ►where and the branches of $s^{-a}$ and ${(1-s)}^{-b}$ are continuous on the path and assume their principal values when $s=c$. … ►

§8.17(vi) Sums

…

17: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

►

§25.11(i) Definition

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

§25.11(ii) Graphics

… ►

§25.11(vi) Derivatives

…

18: 1.10 Functions of a Complex Variable

… ►Also, let $f(z)$ be analytic within $C$, continuous within and on $C$, and real on $\mathrm{𝐴𝐵}$. … ►If $f(z)$ is analytic within a simple closed contour $C$, and continuous within and on $C$—except in both instances for a finite number of singularities within $C$—then … ►Assume that for each $t\in [a,b]$, $f(z,t)$ is an analytic function of $z$ in $D$, and also that $f(z,t)$ is a continuous function of both variables. … ►For each $t\in [a,b)$, $f(z,t)$ is analytic in $D$; $f(z,t)$ is a continuous function of both variables when $z\in D$ and $t\in [a,b)$; the integral (1.10.18) converges at $b$, and this convergence is uniform with respect to $z$ in every compact subset $S$ of $D$. … ►

§1.10(xi) Generating Functions

…

19: 25.1 Special Notation

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

$k,m,n$	nonnegative integers.
…
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 $L(s,\chi )$.

20: 12.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. ►The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …An older notation, due to Whittaker (1902), for $U(a,z)$ is $D_{\nu }\left(z\right)$. …