general%20lemniscatic%20case

SearchAdvancedHelp

(0.003 seconds)

1—10 of 556 matching pages

1: 16.2 Definition and Analytic Properties

… ►

§16.2(i) Generalized Hypergeometric Series

… ►

§16.2(ii) Case $p\le q$

… ►

§16.2(iii) Case $p=q+1$

… ►

§16.2(iv) Case $p>q+1$

►

Polynomials

…

2: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

… ►

§8.19(ii) Graphics

… ►

§8.19(ix) Inequalities

… ►

§8.19(x) Integrals

… ►

§8.19(xi) Further Generalizations

…

3: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

►

§8.21(i) Definitions: General Values

… ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … ►

§8.21(iv) Interrelations

… ►

4: 1.16 Distributions

… ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ (see §1.4(v)) can be considered as a distribution: … ►More generally, if $\alpha (x)$ is an infinitely differentiable function, then … ►Since ${\delta }_{x_0}$ is the Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ corresponding to $\alpha (x)=H\left(x-x_0\right)$ (see §1.4(v)), formula (1.16.16) is a special case of (1.16.3_5), (1.16.9_5) for that choice of $\alpha$. … ►Friedman (1990) gives an overview of generalized functions and their relation to distributions. …

5: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

►

§35.8(i) Definition

… ►

Convergence Properties

… ►

§35.8(iii) ${}_3{}^{}F_2^{}$ Case

… ►

§35.8(iv) General Properties

…

6: 19.2 Definitions

… ►

§19.2(i) General Elliptic Integrals

… ►Also, if $k^2$ and ${\alpha }^2$ are real, then $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is called a circular or hyperbolic case according as ${\alpha }^2({\alpha }^2-k^2)({\alpha }^2-1)$ is negative or positive. … ►The cases with $\varphi =\pi /2$ are the complete integrals: … ►special cases include … ►Formulas involving $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using $R_C(x,y)$. …

7: 20 Theta Functions

Chapter 20 Theta Functions

…

8: 8.26 Tables

… ►

Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

… ►

§8.26(iv) Generalized Exponential Integral

►

Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

… ►

Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

… ►

Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

9: 23 Weierstrass Elliptic and Modular
Functions

…

10: 23.5 Special Lattices

… ►This happens in the cases treated in the following four subsections. … ►

§23.5(iii) Lemniscatic Lattice

… ►

§23.5(iv) Rhombic Lattice

… ► $e_1$ and $g_3$ have the same sign unless $2{\omega }_3=(1+\mathrm{i}){\omega }_1$ when both are zero: the pseudo-lemniscatic case. As a function of $\mathrm{\Im }{e}_3$ the root $e_1$ is increasing. …