# L orthornormal basis

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##### 1: 25.15 Dirichlet $L$-functions
###### §25.15(i) Definitions and Basic Properties
The notation $L\left(s,\chi\right)$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series … …
##### 2: 23.21 Physical Applications
In §22.19(ii) it is noted that Jacobian elliptic functions provide a natural basis of solutions for problems in Newtonian classical dynamics with quartic potentials in canonical form $(1-x^{2})(1-k^{2}x^{2})$. …
23.21.1 $\frac{x^{2}}{\rho-e_{1}}+\frac{y^{2}}{\rho-e_{2}}+\frac{z^{2}}{\rho-e_{3}}=1,$
23.21.5 $\left(\wp\left(v\right)-\wp\left(w\right)\right)\left(\wp\left(w\right)-\wp% \left(u\right)\right)\left(\wp\left(u\right)-\wp\left(v\right)\right)\nabla^{2% }=\left(\wp\left(w\right)-\wp\left(v\right)\right)\frac{{\partial}^{2}}{{% \partial u}^{2}}+\left(\wp\left(u\right)-\wp\left(w\right)\right)\frac{{% \partial}^{2}}{{\partial v}^{2}}+\left(\wp\left(v\right)-\wp\left(u\right)% \right)\frac{{\partial}^{2}}{{\partial w}^{2}}.$
##### 3: 31.15 Stieltjes Polynomials
31.15.12 $\rho(z)=\left(\prod_{j=1}^{N-1}\prod_{k=1}^{N}|z_{j}-a_{k}|^{\gamma_{k}-1}% \right)\left(\prod_{j
The normalized system of products (31.15.8) forms an orthonormal basis in the Hilbert space $L_{\rho}^{2}(Q)$. For further details and for the expansions of analytic functions in this basis see Volkmer (1999).
##### 4: 18.4 Graphics Figure 18.4.5: Laguerre polynomials L ( n ) ⁡ ( x ) , n = 1 , 2 , 3 , 4 , 5 . Magnify Figure 18.4.6: Laguerre polynomials L 3 ( α ) ⁡ ( x ) , α = 0 , 1 , 2 , 3 , 4 . Magnify Figure 18.4.8: Laguerre polynomials L 3 ( α ) ⁡ ( x ) , 0 ≤ α ≤ 3 , 0 ≤ x ≤ 10 . Magnify 3D Help Figure 18.4.9: Laguerre polynomials L 4 ( α ) ⁡ ( x ) , 0 ≤ α ≤ 3 , 0 ≤ x ≤ 10 . Magnify 3D Help
##### 5: 18.41 Tables
Abramowitz and Stegun (1964, Tables 22.4, 22.6, 22.11, and 22.13) tabulates $T_{n}\left(x\right)$, $U_{n}\left(x\right)$, $L^{(n)}\left(x\right)$, and $H_{n}\left(x\right)$ for $n=0(1)12$. The ranges of $x$ are $0.2(.2)1$ for $T_{n}\left(x\right)$ and $U_{n}\left(x\right)$, and $0.5,1,3,5,10$ for $L^{(n)}\left(x\right)$ and $H_{n}\left(x\right)$. … For $P^{n}\left(x\right)$, $L^{(n)}\left(x\right)$, and $H_{n}\left(x\right)$ see §3.5(v). …
##### 6: 30.15 Signal Analysis
The sequence $\phi_{n}$, $n=0,1,2,\dots$ forms an orthonormal basis in the space of $\sigma$-bandlimited functions, and, after normalization, an orthonormal basis in $L^{2}(-\tau,\tau)$. … taken over all $f\in L^{2}(-\infty,\infty)$ subject to …
##### 7: 23.10 Addition Theorems and Other Identities
23.10.10 $\sigma\left(2z\right)=-\wp'\left(z\right){\sigma}^{4}\left(z\right).$
23.10.17 $\wp\left(cz|c\mathbb{L}\right)=c^{-2}\wp\left(z|\mathbb{L}\right),$
23.10.19 $\sigma\left(cz|c\mathbb{L}\right)=c\sigma\left(z|\mathbb{L}\right).$
Also, when $\mathbb{L}$ is replaced by $c\mathbb{L}$ the lattice invariants $g_{2}$ and $g_{3}$ are divided by $c^{4}$ and $c^{6}$, respectively. …
##### 8: 23.14 Integrals
23.14.1 $\int\wp\left(z\right)\mathrm{d}z=-\zeta\left(z\right),$
23.14.2 $\int{\wp}^{2}\left(z\right)\mathrm{d}z=\frac{1}{6}\wp'\left(z\right)+\frac{1}{% 12}g_{2}z,$
23.14.3 $\int{\wp}^{3}\left(z\right)\mathrm{d}z=\frac{1}{120}\wp'''\left(z\right)-\frac% {3}{20}g_{2}\zeta\left(z\right)+\frac{1}{10}g_{3}z.$
##### 9: 23.2 Definitions and Periodic Properties
The generators of a given lattice $\mathbb{L}$ are not unique. …where $a,b,c,d$ are integers, then $2\chi_{1}$, $2\chi_{3}$ are generators of $\mathbb{L}$ iff … When $z\notin\mathbb{L}$ the functions are related by … When it is important to display the lattice with the functions they are denoted by $\wp\left(z|\mathbb{L}\right)$, $\zeta\left(z|\mathbb{L}\right)$, and $\sigma\left(z|\mathbb{L}\right)$, respectively. … If $2\omega_{1}$, $2\omega_{3}$ is any pair of generators of $\mathbb{L}$, and $\omega_{2}$ is defined by (23.2.1), then …
##### 10: 19.33 Triaxial Ellipsoids
The external field and the induced magnetization together produce a uniform field inside the ellipsoid with strength $H/(1+L_{c}\chi)$, where $L_{c}$ is the demagnetizing factor, given in cgs units by
19.33.7 $L_{c}=2\pi abc\int_{0}^{\infty}\frac{\mathrm{d}\lambda}{\sqrt{(a^{2}+\lambda)(% b^{2}+\lambda)(c^{2}+\lambda)^{3}}}=VR_{D}\left(a^{2},b^{2},c^{2}\right).$
where $L_{a}$ and $L_{b}$ are obtained from $L_{c}$ by permutation of $a$, $b$, and $c$. …