functions s(ϵ,ℓ;r),c(ϵ,ℓ;r)

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1: 7.2 Definitions
7.2.8 $S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t,$
$\mathcal{F}\left(z\right)$, $C\left(z\right)$, 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. …
7.2.10 $\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right),$
7.2.11 $\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right).$
3: 28.2 Definitions and Basic Properties
§28.2(iv) Floquet Solutions
28.2.18 $w(z)=\sum_{n=-\infty}^{\infty}c_{2n}e^{\mathrm{i}(\nu+2n)z}$ Figure 28.2.1: Eigenvalues a n ⁡ ( q ) , b n ⁡ ( q ) of Mathieu’s equation as functions of q for 0 ≤ q ≤ 10 , n = 0 , 1 , 2 , 3 , 4 ( a ’s), n = 1 , 2 , 3 , 4 ( b ’s). Magnify
4: 28.20 Definitions and Basic Properties
28.20.2 ${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$ $\zeta=\cosh z$.
28.20.6 $\mathrm{Fe}_{n}\left(z,q\right)=\mp\mathrm{i}\mathrm{fe}_{n}\left(\pm\mathrm{i% }z,q\right),$ $n=0,1,\dots$,
28.20.7 $\mathrm{Ge}_{n}\left(z,q\right)=\mathrm{ge}_{n}\left(\pm\mathrm{i}z,q\right),$ $n=1,2,\dots$.
5: 19.2 Definitions
Let $s^{2}(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. …
19.2.2 $r(s,t)=\frac{(p_{1}+p_{2}s)(p_{3}-p_{4}s)s}{(p_{3}+p_{4}s)(p_{3}-p_{4}s)s}=% \frac{\rho}{s}+\sigma,$
19.2.17 $R_{C}\left(x,y\right)=\frac{1}{2}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t+x% }(t+y)},$
6: 25.1 Special Notation
The main function treated in this chapter is the Riemann zeta function $\zeta\left(s\right)$. … The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\mathrm{Li}_{2}\left(z\right)$, the polylogarithm $\mathrm{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
7: 31.6 Path-Multiplicative Solutions
A further extension of the notation (31.4.1) and (31.4.3) is given by
31.6.1 $(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z% \right),$ $m=0,1,2,\dots$,
8: 4.44 Other Applications
The Einstein functions and Planck’s radiation function are elementary combinations of exponentials, or exponentials and logarithms. …
9: 31.4 Solutions Analytic at Two Singularities: Heun Functions
To emphasize this property this set of functions is denoted by
31.4.1 $(0,1)\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$.
31.4.3 $(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right),$ $m=0,1,2,\dots$,
The set $q_{m}$ depends on the choice of $s_{1}$ and $s_{2}$. …
10: 31.1 Special Notation
The main functions treated in this chapter are $\mathit{H\!\ell}\left(a,q;\alpha,\beta,\gamma,\delta;z\right)$, $(s_{1},s_{2})\mathit{Hf}_{m}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$, $(s_{1},s_{2})\mathit{Hf}_{m}^{\nu}\left(a,q_{m};\alpha,\beta,\gamma,\delta;z\right)$, and the polynomial $\mathit{Hp}_{n,m}\left(a,q_{n,m};-n,\beta,\gamma,\delta;z\right)$. …Sometimes the parameters are suppressed.