# change of variables

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## 1—10 of 93 matching pages

##### 1: 22.7 Landen Transformations
22.7.1 $k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},$
$k_{2}=\frac{2\sqrt{k}}{1+k},$
$k_{2}^{\prime}=\frac{1-k}{1+k},$
22.7.6 $\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{\prime})\operatorname{sn}% \left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{cn}\left(z/(1+k_{2}^{% \prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right% )},$
22.7.8 $\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.$
##### 2: 23.21 Physical Applications
Another form is obtained by identifying $e_{1}$, $e_{2}$, $e_{3}$ as lattice roots (§23.3(i)), and setting …
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: 22.17 Moduli Outside the Interval [0,1]
$k_{1}=\frac{k}{\sqrt{1+k^{2}}},$
$k_{1}k_{1}^{\prime}=\frac{k}{1+k^{2}},$
22.17.6 $\operatorname{sn}\left(z,ik\right)=k_{1}^{\prime}\operatorname{sd}\left(z/k_{1% }^{\prime},k_{1}\right),$
22.17.7 $\operatorname{cn}\left(z,ik\right)=\operatorname{cd}\left(z/k_{1}^{\prime},k_{% 1}\right),$
22.17.8 $\operatorname{dn}\left(z,ik\right)=\operatorname{nd}\left(z/k_{1}^{\prime},k_{% 1}\right).$
$k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},$
$c_{1}={\csc}^{2}\phi_{1}.$
$k_{2}=2\sqrt{k}/(1+k),$
$k_{1}=(1-k^{\prime})/(1+k^{\prime}),$
$c={\csc}^{2}\phi.$
##### 5: 19.7 Connection Formulas
$\Pi\left(\phi,\alpha^{2},k_{1}\right)=k\Pi\left(\beta,k^{2}\alpha^{2},k\right),$ $k_{1}=1/k$, $\sin\beta=k_{1}\sin\phi\leq 1$.
$\kappa=\frac{k}{\sqrt{1+k^{2}}},$
$\kappa^{\prime}=\frac{1}{\sqrt{1+k^{2}}},$
With $\sinh\phi=\tan\psi$, … The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of $\Pi\left(\phi,\alpha^{2},k\right)$ when $\alpha^{2}>{\csc}^{2}\phi$ (see (19.6.5) for the complete case). …
##### 6: 22.11 Fourier and Hyperbolic Series
22.11.1 $\operatorname{sn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.2 $\operatorname{cn}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{q^{n% +\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1+q^{2n+1}},$
22.11.3 $\operatorname{dn}\left(z,k\right)=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{% \infty}\frac{q^{n}\cos\left(2n\zeta\right)}{1+q^{2n}}.$
22.11.4 $\operatorname{cd}\left(z,k\right)=\frac{2\pi}{Kk}\sum_{n=0}^{\infty}\frac{(-1)% ^{n}q^{n+\frac{1}{2}}\cos\left((2n+1)\zeta\right)}{1-q^{2n+1}},$
22.11.5 $\operatorname{sd}\left(z,k\right)=\frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}% \frac{(-1)^{n}q^{n+\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}},$
##### 8: 9.7 Asymptotic Expansions
9.7.1 $\zeta=\tfrac{2}{3}z^{\ifrac{3}{2}}.$
9.7.18 $\operatorname{Ai}\left(z\right)=\frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\left(% \sum_{k=0}^{n-1}(-1)^{k}\frac{u_{k}}{\zeta^{k}}+R_{n}(z)\right),$
9.7.19 $\operatorname{Ai}'\left(z\right)=-\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\left(% \sum_{k=0}^{n-1}(-1)^{k}\frac{v_{k}}{\zeta^{k}}+S_{n}(z)\right),$
9.7.20 $R_{n}(z)=(-1)^{n}\sum_{k=0}^{m-1}(-1)^{k}u_{k}\frac{G_{n-k}\left(2\zeta\right)% }{\zeta^{k}}+R_{m,n}(z),$
9.7.21 $S_{n}(z)=(-1)^{n-1}\sum_{k=0}^{m-1}(-1)^{k}v_{k}\frac{G_{n-k}\left(2\zeta% \right)}{\zeta^{k}}+S_{m,n}(z),$
##### 9: 7.16 Generalized Error Functions
These functions can be expressed in terms of the incomplete gamma function $\gamma\left(a,z\right)$8.2(i)) by change of integration variable.
##### 10: 7.5 Interrelations
7.5.7 $\zeta=\tfrac{1}{2}\sqrt{\pi}(1\mp i)z,$
7.5.9 $C\left(z\right)\pm iS\left(z\right)=\tfrac{1}{2}(1\pm i)\left(1-e^{\pm\frac{1}% {2}\pi iz^{2}}w\left(i\zeta\right)\right).$