# change of parameter

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## 21—30 of 54 matching pages

##### 21: 28.12 Definitions and Basic Properties
For change of signs of $\nu$ and $q$, … For changes of sign of $\nu$, $q$, and $z$, For change of signs of $\nu$ and $z$,
##### 22: 2.5 Mellin Transform Methods
2.5.41 $I_{1}(x)=\mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(1\right)x^{-1}+\frac{% 1}{2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x^{-z}\Gamma\left(1-z\right)% \mathscr{M}\mskip-3.0mu h_{1}\mskip 3.0mu \left(z\right)\mathrm{d}z,$
2.5.42 $I_{2}(x)=\sum_{\Re\beta_{0}\leq\Re z\leq 1}\Residue\left[-x^{-z}\Gamma\left(1-% z\right)\mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)\right]+\frac{% 1}{2\pi i}\int_{\rho-i\infty}^{\rho+i\infty}x^{-z}\Gamma\left(1-z\right)% \mathscr{M}\mskip-3.0mu h_{2}\mskip 3.0mu \left(z\right)\mathrm{d}z.$
##### 23: 1.14 Integral Transforms
1.14.33 $\lim_{t\to\pm\infty}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(\sigma+it% \right)=0.$
1.14.34 $\tfrac{1}{2}(f(u+)+f(u-))=\frac{1}{2\pi i}\lim_{T\to\infty}\int^{\sigma+iT}_{% \sigma-iT}u^{-s}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(s\right)\mathrm{d}s.$
1.14.35 $f(x)=\frac{1}{2\pi i}\int^{\sigma+i\infty}_{\sigma-i\infty}x^{-s}\mathscr{M}% \mskip-3.0mu f\mskip 3.0mu \left(s\right)\mathrm{d}s.$
1.14.36 $\int^{\infty}_{0}f(x)g(yx)\mathrm{d}x=\frac{1}{2\pi i}\*\int^{\sigma+i\infty}_% {\sigma-i\infty}y^{-s}\mathscr{M}\mskip-3.0mu f\mskip 3.0mu \left(1-s\right)% \mathscr{M}\mskip-3.0mu g\mskip 3.0mu \left(s\right)\mathrm{d}s,$
1.14.49 $\lim_{t\to 0+}\frac{\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(-\sigma-it% \right)-\mathcal{S}\mskip-3.0mu f\mskip 3.0mu \left(-\sigma+it\right)}{2\pi i}% =\tfrac{1}{2}(f(\sigma+)+f(\sigma-)),$
##### 24: 19.2 Definitions
19.2.11 $\mathrm{cel}\left(k_{c},p,a,b\right)=\int_{0}^{\pi/2}\frac{a{\cos}^{2}\theta+b% {\sin}^{2}\theta}{{\cos}^{2}\theta+p{\sin}^{2}\theta}\frac{\mathrm{d}\theta}{% \sqrt{{\cos}^{2}\theta+k_{c}^{2}{\sin}^{2}\theta}},$
Here $a,b,p$ are real parameters, and $k_{c}$ and $x$ are real or complex variables, with $p\neq 0$, $k_{c}\neq 0$. …
$k_{c}=k^{\prime},$
$p=1-\alpha^{2},$
$x=\tan\phi,$
##### 25: 11.11 Asymptotic Expansions of Anger–Weber Functions
For fixed $\lambda$ $(>0)$, …
11.11.9_5 $a_{k+1}(\lambda)=\frac{\lambda}{1-\lambda^{2}}\frac{\lambda a^{\prime\prime}_{% k}(\lambda)+a^{\prime}_{k}(\lambda)}{(2k+1)(2k+2)},$ $k=0,1,2,\ldots$.
11.11.12 $\mu=\sqrt{1-\lambda^{2}}-\ln\left(\frac{1+\sqrt{1-\lambda^{2}}}{\lambda}\right),$
11.11.13_5 ${\left(\tfrac{1}{2}\right)_{k}}b_{k}(\lambda)=\frac{(-1)^{k}}{\left(1-\lambda^% {2}\right)^{1/4}}U_{k}\left(\frac{1}{\sqrt{1-\lambda^{2}}}\right),$ $k=0,1,2,\ldots$,
11.11.14 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\frac{1}{\pi\nu(\lambda-1)},$ $\lambda>1$, $|\operatorname{ph}\nu|\leq\pi-\delta$,
##### 26: 12.11 Zeros
###### §12.11(iii) Asymptotic Expansions for Large Parameter
12.11.4 $u_{a,s}\sim 2^{\frac{1}{2}}\mu\left(p_{0}(\alpha)+\frac{p_{1}(\alpha)}{\mu^{4}% }+\frac{p_{2}(\alpha)}{\mu^{8}}+\cdots\right),$
12.11.5 $p_{0}(\zeta)=t(\zeta),$
12.11.6 $p_{1}(\zeta)=\frac{t^{3}-6t}{24(t^{2}-1)^{2}}+\frac{5}{48((t^{2}-1)\zeta^{3})^% {\frac{1}{2}}}.$
12.11.8 $q_{0}(\zeta)=t(\zeta).$
##### 27: 28.4 Fourier Series
28.4.12 $\sum_{m=0}^{\infty}\left(B^{2n+2}_{2m+2}(q)\right)^{2}=1.$
##### 28: 12.14 The Function $W\left(a,x\right)$
These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. …
##### 29: 25.11 Hurwitz Zeta Function
25.11.15 $\zeta\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}\zeta\left(s,a+\frac{n}{k}% \right),$ $s\neq 1$, $k=1,2,3,\dots$.
25.11.31 $\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh x}% \mathrm{d}x=4^{-s}\left(\zeta\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\zeta% \left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right),$ $\Re s>0$, $\Re a>-1$.
25.11.35 $\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x=2^{-s}\left(\zeta% \left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right),$ $\Re a>0$, $\Re s>0$; or $\Re a=0$, $\Im a\neq 0$, $0<\Re s<1$.
##### 30: 20.11 Generalizations and Analogs
###### §20.11(iii) Ramanujan’s Change of Base
20.11.5 ${{}_{2}F_{1}}\left(\tfrac{1}{2},\tfrac{1}{2};1;k^{2}\right)={\theta_{3}}^{2}% \left(0\middle|\tau\right);$
These results are called Ramanujan’s changes of base. …