# modified Mathieu equation

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##### 1: 28.20 Definitions and Basic Properties
###### §28.20(i) ModifiedMathieu’s Equation
When $z$ is replaced by $\pm\mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation:
28.20.2 ${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$ $\zeta=\cosh z$.
28.20.6 $\operatorname{Fe}_{n}\left(z,q\right)=\mp\mathrm{i}\operatorname{fe}_{n}\left(% \pm\mathrm{i}z,q\right),$ $n=0,1,\dots$,
##### 2: 28.23 Expansions in Series of Bessel Functions
28.23.6 ${\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\operatorname{ce}% _{2m}\left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell% }^{2m}(h^{2}){\cal C}_{2\ell}^{(j)}(2h\cosh z),$
28.23.7 ${\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\operatorname{ce}% _{2m}\left(\tfrac{1}{2}\pi,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}A_{2% \ell}^{2m}(h^{2}){\cal C}_{2\ell}^{(j)}(2h\sinh z),$
28.23.8 ${\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\operatorname{% ce}_{2m+1}\left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{% 2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
28.23.10 ${\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\operatorname{% se}_{2m+1}'\left(0,h^{2}\right)\right)^{-1}\tanh z\sum_{\ell=0}^{\infty}(-1)^{% \ell}(2\ell+1)B_{2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
28.23.12 ${\operatorname{Ms}^{(j)}_{2m+2}}\left(z,h\right)=(-1)^{m}\left(\operatorname{% se}_{2m+2}'\left(0,h^{2}\right)\right)^{-1}\tanh z\sum_{\ell=0}^{\infty}(-1)^{% \ell}(2\ell+2)B_{2\ell+2}^{2m+2}(h^{2}){\cal C}_{2\ell+2}^{(j)}(2h\cosh z),$
##### 3: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
28.24.7 $\operatorname{Io}_{2m+2}\left(z,h\right)=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{% \ell}\dfrac{B_{2\ell+2}^{2m+2}(h^{2})}{B_{2s+2}^{2m+2}(h^{2})}\left(I_{\ell-s}% \left(he^{-z}\right)I_{\ell+s+2}\left(he^{z}\right)-I_{\ell+s+2}\left(he^{-z}% \right)I_{\ell-s}\left(he^{z}\right)\right),$
28.24.8 $\operatorname{Ie}_{2m+1}\left(z,h\right)=(-1)^{s}\sum_{\ell=0}^{\infty}(-1)^{% \ell}\dfrac{B_{2\ell+1}^{2m+1}(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}% \left(he^{-z}\right)I_{\ell+s+1}\left(he^{z}\right)+I_{\ell+s+1}\left(he^{-z}% \right)I_{\ell-s}\left(he^{z}\right)\right),$
28.24.11 $\operatorname{Ko}_{2m+2}\left(z,h\right)=\sum_{\ell=0}^{\infty}\frac{B_{2\ell+% 2}^{2m+2}(h^{2})}{B_{2s+2}^{2m+2}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K% _{\ell+s+2}\left(he^{z}\right)-I_{\ell+s+2}\left(he^{-z}\right)K_{\ell-s}\left% (he^{z}\right)\right),$
28.24.12 $\operatorname{Ke}_{2m+1}\left(z,h\right)=\sum_{\ell=0}^{\infty}\frac{B_{2\ell+% 1}^{2m+1}(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K% _{\ell+s+1}\left(he^{z}\right)-I_{\ell+s+1}\left(he^{-z}\right)K_{\ell-s}\left% (he^{z}\right)\right),$
28.24.13 $\operatorname{Ko}_{2m+1}\left(z,h\right)=\sum_{\ell=0}^{\infty}\frac{A_{2\ell+% 1}^{2m+1}(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(I_{\ell-s}\left(he^{-z}\right)K% _{\ell+s+1}\left(he^{z}\right)+I_{\ell+s+1}\left(he^{-z}\right)K_{\ell-s}\left% (he^{z}\right)\right).$
##### 4: 28.22 Connection Formulas
28.22.1 ${\operatorname{Mc}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\dfrac{1}{g% _{\mathit{e},m}(h)\operatorname{ce}_{m}\left(0,h^{2}\right)}\operatorname{Ce}_% {m}\left(z,h^{2}\right),$
28.22.2 ${\operatorname{Ms}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\frac{1}{g_% {\mathit{o},m}(h)\operatorname{se}_{m}'\left(0,h^{2}\right)}\operatorname{Se}_% {m}\left(z,h^{2}\right),$
28.22.4 ${\operatorname{Ms}^{(2)}_{m}}\left(z,h\right)=\sqrt{\frac{2}{\pi}}\dfrac{1}{g_% {\mathit{o},m}(h)\operatorname{se}_{m}'\left(0,h^{2}\right)}\*\left(-f_{% \mathit{o},m}(h)\operatorname{Se}_{m}\left(z,h^{2}\right)-\dfrac{2}{\pi S_{m}(% h^{2})}\operatorname{Ge}_{m}\left(z,h^{2}\right)\right).$
28.22.13 ${\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)=\frac{{\operatorname{M}^{(1)}_{% \nu}}\left(0,h\right)}{\operatorname{me}_{\nu}\left(0,h^{2}\right)}% \operatorname{Me}_{\nu}\left(z,h^{2}\right).$
28.22.14 ${\operatorname{M}^{(2)}_{\nu}}\left(z,h\right)=\cot\left(\nu\pi\right){% \operatorname{M}^{(1)}_{\nu}}\left(z,h\right)-\frac{1}{\sin\left(\nu\pi\right)% }{\operatorname{M}^{(1)}_{-\nu}}\left(z,h\right).$
##### 5: 28.8 Asymptotic Expansions for Large $q$
Barrett (1981) supplies asymptotic approximations for numerically satisfactory pairs of solutions of both Mathieu’s equation (28.2.1) and the modified Mathieu equation (28.20.1). …
##### 6: 28.32 Mathematical Applications
The separated solutions $V(\xi,\eta)=v(\xi)w(\eta)$ can be obtained from the modified Mathieu’s equation (28.20.1) for $v$ and from Mathieu’s equation (28.2.1) for $w$, where $a$ is the separation constant and $q=\tfrac{1}{4}c^{2}k^{2}$. …
##### 7: 28.26 Asymptotic Approximations for Large $q$
28.26.1 ${\operatorname{Mc}^{(3)}_{m}}\left(z,h\right)=\dfrac{e^{\mathrm{i}\phi}}{(\pi h% \cosh z)^{\ifrac{1}{2}}}\*\left(\mathrm{Fc}_{m}\left(z,h\right)-\mathrm{i}% \mathrm{Gc}_{m}\left(z,h\right)\right),$
28.26.2 $\mathrm{i}{\operatorname{Ms}^{(3)}_{m+1}}\left(z,h\right)=\dfrac{e^{\mathrm{i}% \phi}}{(\pi h\cosh z)^{\ifrac{1}{2}}}\*{\left(\mathrm{Fs}_{m}\left(z,h\right)-% \mathrm{i}\mathrm{Gs}_{m}\left(z,h\right)\right)},$
##### 8: 28.25 Asymptotic Expansions for Large $\Re z$
28.25.1 ${\operatorname{M}^{(3,4)}_{\nu}}\left(z,h\right)\sim\frac{e^{\pm\mathrm{i}% \left(2h\cosh z-\left(\frac{1}{2}\nu+\frac{1}{4}\right)\pi\right)}}{\left(\pi h% (\cosh z+1)\right)^{\frac{1}{2}}}\*\sum_{m=0}^{\infty}\dfrac{D^{\pm}_{m}}{% \left(\mp 4\mathrm{i}h(\cosh z+1)\right)^{m}},$
##### 9: 28.35 Tables
###### §28.35 Tables
• Blanch and Clemm (1969) includes eigenvalues $a_{n}\left(q\right)$, $b_{n}\left(q\right)$ for $q=\rho e^{\mathrm{i}\phi}$, $\rho=0(.5)25$, $\phi=5^{\circ}(5^{\circ})90^{\circ}$, $n=0(1)15$; 4D. Also $a_{n}\left(q\right)$ and $b_{n}\left(q\right)$ for $q=\mathrm{i}\rho$, $\rho=0(.5)100$, $n=0(2)14$ and $n=2(2)16$, respectively; 8D. Double points for $n=0(1)15$; 8D. Graphs are included.

• ##### 10: 28.1 Special Notation
The main functions treated in this chapter are the Mathieu functions …and the modified Mathieu functions …The functions ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)$ are also known as the radial Mathieu functions. The eigenvalues of Mathieu’s equation are denoted by … The radial functions ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)$ are denoted by ${\operatorname{Mc}^{(j)}_{n}}\left(z,q\right)$ and ${\operatorname{Ms}^{(j)}_{n}}\left(z,q\right)$, respectively.