# integral equation for Lam� functions

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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 $\sigma$ in $a<\sigma, 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. …
###### §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(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. …
##### 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. …
##### 8: 28.2 Definitions and Basic Properties
###### §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:
##### 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 $p,q\in\mathbb{Z}$, 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 $\wp$ see §23.2(ii). …