integral%20equation%20for%20Lam%C3%A9%20functions

SearchAdvancedHelp

The term"lam" was not found.Possible alternative term: "lame".

(0.008 seconds)

1—10 of 989 matching pages

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 $(0,\mathrm{\infty })$ for all $\sigma$ in , then the integral (1.14.32) converges and $ℳf\left(s\right)$ is an analytic function of $s$ in the vertical strip . … ►

§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 (2000), 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(ii) Dawson’s Integral

… ►

§7.2(iii) Fresnel Integrals

… ► $ℱ\left(z\right)$, $C\left(z\right)$, 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

… ►

§7.2(v) Goodwin–Staton Integral

…

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: 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

… ► $\mathrm{Si}\left(z\right)$ is an odd entire function. … ►

§6.2(iii) Auxiliary Functions

…

6: 19.16 Definitions

… ►

§19.16(i) Symmetric Integrals

… ►Just as the elementary function $R_C(x,y)$ (§19.2(iv)) is the degenerate case … ►

§19.16(ii) $R_{-a}(𝐛;𝐳)$

►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. …

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(viii) Asymptotic Expansions

… ►

8: 19.2 Definitions

… ►

§19.2(i) General Elliptic Integrals

… ►

§19.2(ii) Legendre’s Integrals

… ►The principal values of $K\left(k\right)$ and $E\left(k\right)$ are even functions. … ►

§19.2(iii) Bulirsch’s Integrals

… ►

§19.2(iv) A Related Function: $R_C(x,y)$

…

9: 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 $p,q\in \mathbb{Z}$, and $K$, $K^{\prime }$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime }(={(1-k^2)}^{1/2})$, respectively; see §19.2(ii). … ►

§29.2(ii) Other Forms

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). …

10: 31.2 Differential Equations

§31.2 Differential Equations

►

§31.2(i) Heun’s Equation

►

31.2.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+\frac{ϵ}{z-a}\right)\frac{dw}{dz}+\frac{\alpha \beta z-q}{z(z-1)(z-a)}w=0,$$ $\alpha +\beta +1=\gamma +\delta +ϵ$.

ⓘ

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§29.2(ii), §31.11(iii), §31.12, §31.13, §31.14(i), §31.17(i), §31.18, §31.2(v), §31.2(v), §31.2(v), §31.2(i), §31.3(i), §31.3(i), §31.3(ii), §31.3(ii), §31.3(ii), §31.3(iii), §31.3(iii), §31.4, §31.5, §31.6, §31.7(ii), §31.8, §31.8, §31.8

Permalink:

http://dlmf.nist.gov/31.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(i), §31.2 and Ch.31

… ►

§31.2(v) Heun’s Equation Automorphisms

… ►

Composite Transformations

…