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1: 1.14 Integral Transforms

§1.14 Integral Transforms

… ►where the last integral denotes the Cauchy principal value (1.4.25). … ►If

x^{\sigma-1}f(x)

is integrable on

(0,\infty)

for all

\sigma

a<\sigma<b

, then the integral (1.14.32) converges and

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(s\right)

is an analytic function of

s

in the vertical strip

a<\Re s<b

. … ►

§1.14(viii) Compendia

►For more extensive tables of the integral transforms of this section and tables of other integral transforms, see Erdélyi et al. (1954a, b), Gradshteyn and Ryzhik (2015), Marichev (1983), Oberhettinger (1972, 1974, 1990), Oberhettinger and Badii (1973), Oberhettinger and Higgins (1961), Prudnikov et al. (1986a, b, 1990, 1992a, 1992b).

2: 7.2 Definitions

… ►

§7.2(i) Error Functions

… ►

§7.2(ii) Dawson’s Integral

… ►

§7.2(iii) Fresnel Integrals

… ►

\mathcal{F}\left(z\right)

C\left(z\right)

, and

S\left(z\right)

are entire functions of

z

, as are

\mathrm{f}\left(z\right)

and

\mathrm{g}\left(z\right)

in the next subsection. … ►

§7.2(iv) Auxiliary Functions

…

3: 8.19 Generalized Exponential Integral

§8.19 Generalized Exponential Integral

►

§8.19(i) Definition and Integral Representations

… ►

Other Integral Representations

… ►

§8.19(vi) Relation to Confluent Hypergeometric Function

… ►

§8.19(x) Integrals

…

4: 7.18 Repeated Integrals of the Complementary Error Function

§7.18 Repeated Integrals of the Complementary Error Function

►

§7.18(i) Definition

… ►

§7.18(iii) Properties

… ► … ►

Hermite Polynomials

…

5: 19.16 Definitions

… ►

§19.16(i) Symmetric Integrals

… ►Just as the elementary function

R_{C}\left(x,y\right)

(§19.2(iv)) is the degenerate case … ►

§19.16(ii) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$

►All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The

R

-function is often used to make a unified statement of a property of several elliptic integrals. …

6: 6.2 Definitions and Interrelations

… ►

§6.2(i) Exponential and Logarithmic Integrals

… ►The logarithmic integral is defined by … ►

§6.2(ii) Sine and Cosine Integrals

… ►

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

is an odd entire function. … ►

§6.2(iii) Auxiliary Functions

…

7: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►

§8.21(iii) Integral Representations

… ►

Spherical-Bessel-Function Expansions

… ►

§8.21(vii) Auxiliary Functions

… ►

§8.21(viii) Asymptotic Expansions

…

8: 28.2 Definitions and Basic Properties

… ►

28.2.1

w^{\prime\prime}+(a-2q\cos\left(2z\right))w=0.

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $q=h^{2}$ : parameter, $z$ : complex variable, $a$ : parameter and $w(z)$ : Mathieu’s equation solution
A&S Ref:: 20.3.1 20.1.1 (in slightly different form)
Referenced by:: §1.18(vii), Figure 28.17.1, Figure 28.17.1, §28.17, §28.17, §28.2(ii), §28.2(ii), §28.2(iii), §28.2(iv), §28.20(i), §28.29(i), §28.32(i), §28.32(i), 1st item, 2nd item, §28.33(iii), item (a), item (c), item (c), §28.5(i), §28.8(iv), §28.8(iv), §28.8(iv), §30.2(iii)
Permalink:: http://dlmf.nist.gov/28.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.2(i), §28.2 and Ch.28

… ►

§28.2(ii) Basic Solutions $w_{\mbox{\rm\tiny I}}$ , $w_{\mbox{\rm\tiny II}}$

… ►

§28.2(iv) Floquet Solutions

… ►

§28.2(vi) Eigenfunctions

… ►For simple roots

q

of the corresponding equations (28.2.21) and (28.2.22), the functions are made unique by the normalizations …

9: 28.20 Definitions and Basic Properties

… ►

§28.20(i) Modified Mathieu’s Equation

►When

z

is replaced by

\pm\mathrm{i}z

, (28.2.1) becomes the modified Mathieu’s equation: ►

28.20.1

w^{\prime\prime}-\left(a-2q\cosh\left(2z\right)\right)w=0,

ⓘ

Symbols:: $\cosh\NVar{z}$ : hyperbolic cosine function, $q=h^{2}$ : parameter, $z$ : complex variable, $a$ : parameter and $w(z)$ : Mathieu’s equation solution
A&S Ref:: 20.1.2 (in slightly different form) 20.8.6
Referenced by:: §28.20(iii), §28.32(i), 1st item, 2nd item, item (b), §28.8(iv), §28.8(iv)
Permalink:: http://dlmf.nist.gov/28.20.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.20(i), §28.20 and Ch.28

… ►

§28.20(iv) Radial Mathieu Functions ${\operatorname{Mc}^{(j)}_{n}}$ , ${\operatorname{Ms}^{(j)}_{n}}$

… ►

§28.20(vi) Wronskians

…

10: 29.2 Differential Equations

… ►

§29.2(i) Lamé’s Equation

… ►This equation has regular singularities at the points

2pK+(2q+1)\mathrm{i}{K^{\prime}}

, where

p,q\in\mathbb{Z}