# change of parameter

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## 1—10 of 54 matching pages

##### 1: 19.15 Advantages of Symmetry
(19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …
##### 2: 19.7 Connection Formulas
###### §19.7(iii) Change of Parameter of $\Pi\left(\phi,\alpha^{2},k\right)$
19.7.10 $(1-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)+(1-\omega^{2})\Pi\left(\phi,% \omega^{2},k\right)=F\left(\phi,k\right)+(1-\alpha^{2}-\omega^{2})\sqrt{c-k^{2% }}\*R_{C}\left(c(c-1),(c-\alpha^{2})(c-\omega^{2})\right),$ $(k^{2}-\alpha^{2})(k^{2}-\omega^{2})=k^{2}(k^{2}-1)$.
##### 3: 19.21 Connection Formulas
###### §19.21(iii) Change of Parameter of $R_{J}$
Change-of-parameter relations can be used to shift the parameter $p$ of $R_{J}$ from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). …
##### 5: 29.6 Fourier Series
29.6.1 $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{% \infty}A_{2p}\cos\left(2p\phi\right).$
29.6.8 $\mathit{Ec}^{2m}_{\nu}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{\infty}C_{2p}\cos\left(2p\phi\right)\right).$
29.6.16 $\mathit{Ec}^{2m+1}_{\nu}\left(z,k^{2}\right)=\sum_{p=0}^{\infty}A_{2p+1}\cos% \left((2p+1)\phi\right).$
29.6.31 $\mathit{Es}^{2m+1}_{\nu}\left(z,k^{2}\right)=\sum_{p=0}^{\infty}B_{2p+1}\sin% \left((2p+1)\phi\right).$
29.6.46 $\mathit{Es}^{2m+2}_{\nu}\left(z,k^{2}\right)=\sum_{p=1}^{\infty}B_{2p}\sin% \left(2p\phi\right).$
##### 6: 29.15 Fourier Series and Chebyshev Series
29.15.1 $\mathit{uE}^{m}_{2n}\left(z,k^{2}\right)=\tfrac{1}{2}A_{0}+\sum_{p=1}^{n}A_{2p% }\cos\left(2p\phi\right).$
29.15.8 $\mathit{sE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}A_{2p+1}\cos\left((2p% +1)\phi\right).$
29.15.13 $\mathit{cE}^{m}_{2n+1}\left(z,k^{2}\right)=\sum_{p=0}^{n}B_{2p+1}\sin\left((2p% +1)\phi\right).$
29.15.18 $\mathit{dE}^{m}_{2n+1}\left(z,k^{2}\right)=\operatorname{dn}\left(z,k\right)% \left(\tfrac{1}{2}C_{0}+\sum_{p=1}^{n}C_{2p}\cos\left(2p\phi\right)\right).$
29.15.23 $\mathit{scE}^{m}_{2n+2}\left(z,k^{2}\right)=\sum_{p=0}^{n}B_{2p+2}\sin\left((2% p+2)\phi\right).$
##### 7: 15.12 Asymptotic Approximations
15.12.5 $\mathbf{F}\left({a+\lambda,b-\lambda\atop c};\tfrac{1}{2}-\tfrac{1}{2}z\right)% =2^{(a+b-1)/2}\frac{(z+1)^{(c-a-b-1)/2}}{(z-1)^{c/2}}\sqrt{\zeta\sinh\zeta}% \left(\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b\right)^{1-c}\left(I_{c-1}\left((% \lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)(1+O(\lambda^{-2}))+\frac{I_{c% -2}\left((\lambda+\tfrac{1}{2}a-\tfrac{1}{2}b)\zeta\right)}{2\lambda+a-b}\left% (\left(c-\tfrac{1}{2}\right)\left(c-\tfrac{3}{2}\right)\left(\frac{1}{\zeta}-% \coth\zeta\right)+\tfrac{1}{2}(2c-a-b-1)(a+b-1)\tanh\left(\tfrac{1}{2}\zeta% \right)+O(\lambda^{-2})\right)\right),$
15.12.9 $(z+1)^{3\lambda/2}(2\lambda)^{c-1}\mathbf{F}\left({a+\lambda,b+2\lambda\atop c% };-z\right)={\lambda^{-1/3}\left({\mathrm{e}}^{\pi\mathrm{i}(a-c+\lambda+(1/3)% )}\mathrm{Ai}\left({\mathrm{e}}^{-\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}% {3}}\beta^{2}\right)+{\mathrm{e}}^{\pi\mathrm{i}(c-a-\lambda-(1/3))}\mathrm{Ai% }\left({\mathrm{e}}^{\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}% \right)\right)\left(a_{0}(\zeta)+O(\lambda^{-1})\right)}+\lambda^{-2/3}\left({% \mathrm{e}}^{\pi\mathrm{i}(a-c+\lambda+(2/3))}\mathrm{Ai}'\left({\mathrm{e}}^{% -\ifrac{2\pi\mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)+{\mathrm{e}}% ^{\pi\mathrm{i}(c-a-\lambda-(2/3))}\mathrm{Ai}'\left({\mathrm{e}}^{\ifrac{2\pi% \mathrm{i}}{3}}\lambda^{\ifrac{2}{3}}\beta^{2}\right)\right)\left(a_{1}(\zeta)% +O(\lambda^{-1})\right),$
15.12.13 $G_{0}(\pm\beta)=\left(2+{\mathrm{e}}^{\pm\zeta}\right)^{c-b-(\ifrac{1}{2})}% \left(1+{\mathrm{e}}^{\pm\zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-{\mathrm{% e}}^{\pm\zeta}\right)^{-a+(\ifrac{1}{2})}\sqrt{\frac{\beta}{{\mathrm{e}}^{% \zeta}-{\mathrm{e}}^{-\zeta}}}.$
##### 8: 30.2 Differential Equations
With $\zeta=\gamma z$ Equation (30.2.1) changes to …
##### 9: 31.2 Differential Equations
31.2.8 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}\zeta}^{2}}+\left((2\gamma-1)\frac{% \operatorname{cn}\zeta\operatorname{dn}\zeta}{\operatorname{sn}\zeta}-(2\delta% -1)\frac{\operatorname{sn}\zeta\operatorname{dn}\zeta}{\operatorname{cn}\zeta}% -(2\epsilon-1)k^{2}\frac{\operatorname{sn}\zeta\operatorname{cn}\zeta}{% \operatorname{dn}\zeta}\right)\frac{\mathrm{d}w}{\mathrm{d}\zeta}+4k^{2}(% \alpha\beta{\operatorname{sn}}^{2}\zeta-q)w=0.$
##### 10: 2.8 Differential Equations with a Parameter
2.8.3 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=\left(u^{2}\dot{z}^{2}f(z)+\psi(% \xi)\right)W,$
2.8.9 $\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}\xi}^{2}}=\left(\frac{u^{2}}{\xi}+\frac{% \rho}{\xi^{2}}\right)W,$