Green theorem for vector-valued functions

SearchAdvancedHelp

(0.005 seconds)

11—20 of 961 matching pages

11: 20.2 Definitions and Periodic Properties

… ►

§20.2(i) Fourier Series

… ►

§20.2(ii) Periodicity and Quasi-Periodicity

… ►The theta functions are quasi-periodic on the lattice: … ►

§20.2(iii) Translation of the Argument by Half-Periods

… ►

§20.2(iv) $z$-Zeros

…

12: 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

… ► …

13: 5.12 Beta Function

§5.12 Beta Function

… ►

Euler’s Beta Integral

… ►

►►

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

… ►

►►

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

►

Pochhammer’s Integral

…

14: 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

…

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: 4.2 Definitions

… ► … ►

§4.2(iii) The Exponential Function

… ►

§4.2(iv) Powers

►

Powers with General Bases

… ► …

17: 1.10 Functions of a Complex Variable

… ►

Picard’s Theorem

… ►

§1.10(iv) Residue Theorem

… ►

Lagrange Inversion Theorem

… ►

Extended Inversion Theorem

… ►

§1.10(xi) Generating Functions

…

18: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ► … ►Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … ►

Polynomials

… ►

§16.2(v) Behavior with Respect to Parameters

…

19: 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

…

20: 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 $L(s,\chi )$.