# change of amplitude

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## 6 matching pages

##### 2: 29.2 Differential Equations
29.2.5 $\phi=\tfrac{1}{2}\pi-\operatorname{am}\left(z,k\right).$
##### 3: 29.1 Special Notation
where $\psi=\operatorname{am}\left(z,k\right)$; see §22.16(i). The relation to the Lamé functions ${\rm Ec}^{m}_{\nu}$, ${\rm Es}^{m}_{\nu}$ of Ince (1940b) is given by …
##### 4: 29.6 Fourier Series
With $\phi=\frac{1}{2}\pi-\operatorname{am}\left(z,k\right)$, as in (29.2.5), we have …
##### 6: 19.25 Relations to Other Functions
then the five nontrivial permutations of $x,y,z$ that leave $R_{F}$ invariant change $k^{2}$ ($=(z-y)/(z-x)$) into $1/k^{2}$, ${k^{\prime}}^{2}$, $1/{k^{\prime}}^{2}$, $-k^{2}/{k^{\prime}}^{2}$, $-{k^{\prime}}^{2}/k^{2}$, and $\sin\phi$ ($=\sqrt{(z-x)/z}$) into $k\sin\phi$, $-i\tan\phi$, $-ik^{\prime}\tan\phi$, $(k^{\prime}\sin\phi)/\sqrt{1-k^{2}{\sin}^{2}\phi}$, $-ik\sin\phi/\sqrt{1-k^{2}{\sin}^{2}\phi}$. … The three changes of parameter of $\Pi\left(\phi,\alpha^{2},k\right)$ in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). …
19.25.30 $\operatorname{am}\left(u,k\right)=R_{C}\left({\operatorname{cs}}^{2}\left(u,k% \right),{\operatorname{ns}}^{2}\left(u,k\right)\right),$
The sign on the right-hand side of (19.25.35) will change whenever one crosses a curve on which $\wp\left(z\right)-e_{j}<0$, for some $j$. … The sign on the right-hand side of (19.25.40) will change whenever one crosses a curve on which $\sigma_{j}^{2}(z)<0$, for some $j$. …