trigonometric expansion

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2: 14.13 Trigonometric Expansions
§14.13 TrigonometricExpansions
For other trigonometric expansions see Erdélyi et al. (1953a, pp. 146–147).
3: 28.26 Asymptotic Approximations for Large $q$
28.26.4 $\mathrm{Fc}_{m}\left(z,h\right)\sim 1+\dfrac{s}{8h{\cosh}^{2}z}+\dfrac{1}{2^{1% 1}h^{2}}\left(\dfrac{s^{4}+86s^{2}+105}{{\cosh}^{4}z}-\dfrac{s^{4}+22s^{2}+57}% {{\cosh}^{2}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(-\dfrac{s^{5}+14s^{3}+33s}{{% \cosh}^{2}z}-\dfrac{2s^{5}+124s^{3}+1122s}{{\cosh}^{4}z}+\dfrac{3s^{5}+290s^{3% }+1627s}{{\cosh}^{6}z}\right)+\cdots,$
28.26.5 $\mathrm{Gc}_{m}\left(z,h\right)\sim\dfrac{\sinh z}{{\cosh}^{2}z}\left(\dfrac{s% ^{2}+3}{2^{5}h}+\dfrac{1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cosh}% ^{2}z}\right)+\dfrac{1}{2^{14}h^{3}}\left(5s^{4}+34s^{2}+9-\dfrac{s^{6}-47s^{4% }+667s^{2}+2835}{12{\cosh}^{2}z}+\dfrac{s^{6}+505s^{4}+12139s^{2}+10395}{12{% \cosh}^{4}z}\right)\right)+\cdots.$
4: 10.19 Asymptotic Expansions for Large Order
$J_{\nu}\left(\nu\operatorname{sech}\alpha\right)\sim\frac{e^{\nu(\tanh\alpha-% \alpha)}}{(2\pi\nu\tanh\alpha)^{\frac{1}{2}}}\sum_{k=0}^{\infty}\frac{U_{k}(% \coth\alpha)}{\nu^{k}},$
$Y_{\nu}\left(\nu\operatorname{sech}\alpha\right)\sim-\frac{e^{\nu(\alpha-\tanh% \alpha)}}{(\tfrac{1}{2}\pi\nu\tanh\alpha)^{\frac{1}{2}}}\*\sum_{k=0}^{\infty}(% -1)^{k}\frac{U_{k}(\coth\alpha)}{\nu^{k}},$
$J_{\nu}'\left(\nu\operatorname{sech}\alpha\right)\sim\left(\frac{\sinh\left(2% \alpha\right)}{4\pi\nu}\right)^{\frac{1}{2}}e^{\nu(\tanh\alpha-\alpha)}\sum_{k% =0}^{\infty}\frac{V_{k}(\coth\alpha)}{\nu^{k}},$
$Y_{\nu}'\left(\nu\operatorname{sech}\alpha\right)\sim\left(\frac{\sinh\left(2% \alpha\right)}{\pi\nu}\right)^{\frac{1}{2}}e^{\nu(\alpha-\tanh\alpha)}\sum_{k=% 0}^{\infty}(-1)^{k}\frac{V_{k}(\coth\alpha)}{\nu^{k}}.$
$J_{\nu}'\left(\nu\sec\beta\right)\sim\left(\frac{\sin\left(2\beta\right)}{\pi% \nu}\right)^{\frac{1}{2}}\*\left(-\sin\xi\sum_{k=0}^{\infty}\frac{V_{2k}(i\cot% \beta)}{\nu^{2k}}-i\cos\xi\sum_{k=0}^{\infty}\frac{V_{2k+1}(i\cot\beta)}{\nu^{% 2k+1}}\right),$
5: 28.8 Asymptotic Expansions for Large $q$
28.8.11 $P_{m}(x)\sim 1+\dfrac{s}{2^{3}h{\cos}^{2}x}+\dfrac{1}{h^{2}}\left(\dfrac{s^{4}% +86s^{2}+105}{2^{11}{\cos}^{4}x}-\dfrac{s^{4}+22s^{2}+57}{2^{11}{\cos}^{2}x}% \right)+\cdots,$
28.8.12 $Q_{m}(x)\sim\dfrac{\sin x}{{\cos}^{2}x}\left(\dfrac{1}{2^{5}h}(s^{2}+3)+\dfrac% {1}{2^{9}h^{2}}\left(s^{3}+3s+\dfrac{4s^{3}+44s}{{\cos}^{2}x}\right)\right)+\cdots.$
6: 6.12 Asymptotic Expansions
The asymptotic expansions of $\operatorname{Si}\left(z\right)$ and $\operatorname{Ci}\left(z\right)$ are given by (6.2.19), (6.2.20), together with When $\frac{1}{4}\pi\leq|\operatorname{ph}z|<\frac{1}{2}\pi$ the remainders are bounded in magnitude by $\csc\left(2|\operatorname{ph}z|\right)$ times the first neglected terms. …
7: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
22.12.13 $2K\operatorname{cs}\left(2Kt,k\right)=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)^{n}% \frac{\pi}{\tan\left(\pi(t-n\tau)\right)}=\lim_{N\to\infty}\sum_{n=-N}^{N}(-1)% ^{n}\left(\lim_{M\to\infty}\sum_{m=-M}^{M}\frac{1}{t-m-n\tau}\right).$
9: 12.10 Uniform Asymptotic Expansions for Large Parameter
12.10.18 $U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\frac{2g(\mu)}{(1-t^{2})^{% \frac{1}{4}}}\*\left(\cos\kappa\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{% \cal{A}}_{2s}(t)}{\mu^{4s}}-\sin\kappa\sum_{s=0}^{\infty}(-1)^{s}\frac{% \widetilde{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),$
12.10.19 $U'\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\mu\sqrt{2}g(\mu)(1-t^{2}% )^{\frac{1}{4}}\*\left(\sin\kappa\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{% \cal{B}}_{2s}(t)}{\mu^{4s}}+\cos\kappa\sum_{s=0}^{\infty}(-1)^{s}\frac{% \widetilde{\cal{B}}_{2s+1}(t)}{\mu^{4s+2}}\right),$
12.10.20 $V\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\frac{2g(\mu)}{\Gamma\left% (\tfrac{1}{2}+\tfrac{1}{2}\mu^{2}\right)(1-t^{2})^{\frac{1}{4}}}\*\left(\cos% \chi\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{A}}_{2s}(t)}{\mu^{4s}}-% \sin\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{A}}_{2s+1}(t)}{\mu^{4% s+2}}\right),$
12.10.21 $V'\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\frac{\mu\sqrt{2}g(\mu)(1% -t^{2})^{\frac{1}{4}}}{\Gamma\left(\tfrac{1}{2}+\tfrac{1}{2}\mu^{2}\right)}\*% \left(\sin\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{B}}_{2s}(t)}{% \mu^{4s}}+\cos\chi\sum_{s=0}^{\infty}(-1)^{s}\frac{\widetilde{\cal{B}}_{2s+1}(% t)}{\mu^{4s+2}}\right),$
10: 12.14 The Function $W\left(a,x\right)$
Other expansions, involving $\cos\left(\tfrac{1}{4}x^{2}\right)$ and $\sin\left(\tfrac{1}{4}x^{2}\right)$, can be obtained from (12.4.3) to (12.4.6) by replacing $a$ by $-ia$ and $z$ by $xe^{\ifrac{\pi i}{4}}$; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16). …
12.14.25 $W\left(\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\frac{2^{-\frac{1}{2}}e^{-% \frac{1}{4}\pi\mu^{2}}l(\mu)}{(t^{2}-1)^{\frac{1}{4}}}\left(\cos\sigma\sum_{s=% 0}^{\infty}(-1)^{s}\frac{{\cal{A}}_{2s}(t)}{\mu^{4s}}-\sin\sigma\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),$
12.14.26 $W\left(\tfrac{1}{2}\mu^{2},-\mu t\sqrt{2}\right)\sim\frac{2^{\frac{1}{2}}e^{% \frac{1}{4}\pi\mu^{2}}l(\mu)}{(t^{2}-1)^{\frac{1}{4}}}\left(\sin\sigma\sum_{s=% 0}^{\infty}(-1)^{s}\frac{{\cal{A}}_{2s}(t)}{\mu^{4s}}+\cos\sigma\sum_{s=0}^{% \infty}(-1)^{s}\frac{{\cal{A}}_{2s+1}(t)}{\mu^{4s+2}}\right),$
12.14.34 $W\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim\frac{l(\mu)}{(t^{2}+1)^{% \frac{1}{4}}}\left(\cos\overline{\sigma}\sum_{s=0}^{\infty}\frac{(-1)^{s}% \overline{u}_{2s}(t)}{(t^{2}+1)^{3s}\mu^{4s}}-\sin\overline{\sigma}\sum_{s=0}^% {\infty}\frac{(-1)^{s}\overline{u}_{2s+1}(t)}{(t^{2}+1)^{3s+\frac{3}{2}}\mu^{4% s+2}}\right),$
12.14.35 $W'\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)\sim-\frac{\mu}{\sqrt{2}}l(% \mu)(t^{2}+1)^{\frac{1}{4}}\left(\sin\overline{\sigma}\sum_{s=0}^{\infty}\frac% {(-1)^{s}\overline{v}_{2s}(t)}{(t^{2}+1)^{3s}\mu^{4s}}+\cos\overline{\sigma}% \sum_{s=0}^{\infty}\frac{(-1)^{s}\overline{v}_{2s+1}(t)}{(t^{2}+1)^{3s+\frac{3% }{2}}\mu^{4s+2}}\right),$