# trigonometric series expansions

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##### 2: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
###### §22.12 Expansions in Other TrigonometricSeries and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.13 $2K\operatorname{cs}\left(2Kt,k\right)=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}% \frac{\pi}{\tan\left(\pi(t-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)% ^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right).$
##### 4: 3.11 Approximation Techniques
In fact, (3.11.11) is the Fourier-series expansion of $f(\cos\theta)$; compare (3.11.6) and §1.8(i). …
##### 5: 6.20 Approximations
• Luke and Wimp (1963) covers $\operatorname{Ei}\left(x\right)$ for $x\leq-4$ (20D), and $\operatorname{Si}\left(x\right)$ and $\operatorname{Ci}\left(x\right)$ for $x\geq 4$ (20D).

• Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\operatorname{Ein}\left(ax\right)$, $\operatorname{Si}\left(ax\right)$, and $\operatorname{Cin}\left(ax\right)$ for $-1\leq x\leq 1$, $a\in\mathbb{C}$. The coefficients are given in terms of series of Bessel functions.

• Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric $U$-function (§13.2(i)) from which Chebyshev expansions near infinity for $E_{1}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the $U$ functions. If $|\operatorname{ph}z|<\pi$ the scheme can be used in backward direction.

• ##### 6: 6.18 Methods of Computation
Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. …
##### 7: 8.21 Generalized Sine and Cosine Integrals
###### Spherical-Bessel-Function Expansions
For (8.21.16), (8.21.17), and further expansions in series of Bessel functions see Luke (1969b, pp. 56–57). … For the corresponding expansions for $\operatorname{si}\left(a,z\right)$ and $\operatorname{ci}\left(a,z\right)$ apply (8.21.20) and (8.21.21). …
##### 8: 19.5 Maclaurin and Related Expansions
###### §19.5 Maclaurin and Related Expansions
Series expansions of $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ are surveyed and improved in Van de Vel (1969), and the case of $F\left(\phi,k\right)$ is summarized in Gautschi (1975, §1.3.2). For series expansions of $\Pi\left(\phi,\alpha^{2},k\right)$ when $|\alpha^{2}|<1$ see Erdélyi et al. (1953b, §13.6(9)). …
##### 9: 28.23 Expansions in Series of Bessel Functions
###### §28.23 Expansions in Series of Bessel Functions
valid for all $z$ when $j=1$, and for $\Re z>0$ and $|\cosh z|>1$ when $j=2,3,4$. … When $j=1$, each of the series (28.23.6)–(28.23.13) converges for all $z$. When $j=2,3,4$ the series in the even-numbered equations converge for $\Re z>0$ and $|\cosh z|>1$, and the series in the odd-numbered equations converge for $\Re z>0$ and $|\sinh z|>1$. …
##### 10: 24.8 Series Expansions
24.8.9 $E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh\left(\tfrac{1}{2}\pi k% \right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1},$ $n=1,2,\dots$.