generalizations

SearchAdvancedHelp

(0.001 seconds)

41—50 of 394 matching pages

41: 4.2 Definitions

… ►

§4.2(i) The Logarithm

… ►With this definition the general logarithm is given by … ►

§4.2(ii) Logarithms to a General Base $a$

… ►The general value of the phase is given by … ►

Powers with General Bases

…

42: 3 Numerical Methods

…

43: 20 Theta Functions

…

44: 21.8 Abelian Functions

… ►In consequence, Abelian functions are generalizations of elliptic functions (§23.2(iii)) to more than one complex variable. …

45: Adri B. Olde Daalhuis

… ►

16 Generalized Hypergeometric Functions & Meijer G-Function

46: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►

§8.20(i) Large $z$

►

8.20.1 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{{\left(p\right)}_k}{z^k}+{(-1)}^n\frac{{\left(p\right)}_n{\mathrm{e}}^z}{z^{n-1}}{E}_{n+p}\left(z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.51

Permalink:

http://dlmf.nist.gov/8.20.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

… ►

8.20.2 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(p\right)}_k}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). … ►

§8.20(ii) Large $p$

…

47: 14.4 Graphics

…

48: 21.10 Methods of Computation

… ►

§21.10(i) General Riemann Theta Functions

…

49: Bonita V. Saunders

… ►Her research interests include numerical grid generation, numerical solution of partial differential equations, and visualization of special functions. … ►She has also used her work for another passion: inspiring the next generation of mathematical scientists with presentations at middle schools, high schools, colleges, and community centers.

50: 8.27 Approximations

… ►

§8.27(ii) Generalized Exponential Integral

►

Luke (1975, p. 103) gives Chebyshev-series expansions for $E_1\left(x\right)$ and related functions for $x\ge 5$.

…