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##### 1: 22.17 Moduli Outside the Interval [0,1]
###### §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 $\operatorname{sn}\left(z,ik\right)=k_{1}^{\prime}\operatorname{sd}\left(z/k_{1% }^{\prime},k_{1}\right),$
For proofs of these results and further information see Walker (2003).
##### 2: 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 $\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{\prime})({\operatorname{dn}}% ^{2}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^{\prime})}{k_{2}^{2}% \operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}.$
##### 3: Sidebar 5.SB1: Gamma & Digamma Phase Plots
In the upper half of the image, the poles of $\Gamma\left(z\right)$ are clearly visible at negative integer values of $z$: the phase changes by $2\pi$ around each pole, showing a full revolution of the color wheel. … Phase changes around the zeros are of opposite sign to those around the poles. …
##### 4: 19.7 Connection Formulas
###### §19.7(ii) Change of Modulus and Amplitude
$\kappa=\frac{k}{\sqrt{1+k^{2}}},$
$\kappa^{\prime}=\frac{1}{\sqrt{1+k^{2}}},$
With $\sinh\phi=\tan\psi$, …
###### §19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$
$k_{1}=\frac{1-k^{\prime}}{1+k^{\prime}},$
$c_{1}={\csc}^{2}\phi_{1}.$
$k_{2}=2\sqrt{k}/(1+k),$
$k_{1}=(1-k^{\prime})/(1+k^{\prime}),$
$c={\csc}^{2}\phi.$
##### 6: 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.
##### 8: 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 $\left(\wp\left(v\right)-\wp\left(w\right)\right)\left(\wp\left(w\right)-\wp% \left(u\right)\right)\left(\wp\left(u\right)-\wp\left(v\right)\right)\nabla^{2% }=\left(\wp\left(w\right)-\wp\left(v\right)\right)\frac{{\partial}^{2}}{{% \partial u}^{2}}+\left(\wp\left(u\right)-\wp\left(w\right)\right)\frac{{% \partial}^{2}}{{\partial v}^{2}}+\left(\wp\left(v\right)-\wp\left(u\right)% \right)\frac{{\partial}^{2}}{{\partial w}^{2}}.$
##### 9: Foreword
Much has changed in the years since A&S was published. …However, we have also seen the birth of a new age of computing technology, which has not only changed how we utilize special functions, but also how we communicate technical information. …
##### 10: 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 $\operatorname{sd}\left(z,k\right)=\frac{2\pi}{Kkk^{\prime}}\sum_{n=0}^{\infty}% \frac{(-1)^{n}q^{n+\frac{1}{2}}\sin\left((2n+1)\zeta\right)}{1+q^{2n+1}},$