# modified Mathieu functions

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##### 1: 28.20 Definitions and Basic Properties
###### §28.20(ii) Solutions $\operatorname{Ce}_{\nu}$, $\operatorname{Se}_{\nu}$, $\operatorname{Me}_{\nu}$, $\operatorname{Fe}_{n}$, $\operatorname{Ge}_{n}$
For other values of $z$, $h$, and $\nu$ the functions ${\operatorname{M}^{(j)}_{\nu}}\left(z,h\right)$, $j=1,2,3,4$, are determined by analytic continuation. …
##### 3: 28.22 Connection Formulas
###### §28.22 Connection Formulas
28.22.4 ${\operatorname{Ms}^{(2)}_{m}}\left(z,h\right)=\sqrt{\frac{2}{\pi}}\dfrac{1}{g_% {\mathit{o},m}(h)\operatorname{se}_{m}'\left(0,h^{2}\right)}\*\left(-f_{% \mathit{o},m}(h)\operatorname{Se}_{m}\left(z,h^{2}\right)-\dfrac{2}{\pi S_{m}(% h^{2})}\operatorname{Ge}_{m}\left(z,h^{2}\right)\right).$
The joining factors in the above formulas are given by …
28.22.13 ${\operatorname{M}^{(1)}_{\nu}}\left(z,h\right)=\frac{{\operatorname{M}^{(1)}_{% \nu}}\left(0,h\right)}{\operatorname{me}_{\nu}\left(0,h^{2}\right)}% \operatorname{Me}_{\nu}\left(z,h^{2}\right).$
##### 4: 28.23 Expansions in Series of Bessel Functions
###### §28.23 Expansions in Series of Bessel Functions
28.23.2 $\operatorname{me}_{\nu}\left(0,h^{2}\right){\operatorname{M}^{(j)}_{\nu}}\left% (z,h\right)=\sum_{n=-\infty}^{\infty}(-1)^{n}c_{2n}^{\nu}(h^{2}){\cal C}_{\nu+% 2n}^{(j)}(2h\cosh z),$
28.23.6 ${\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\operatorname{ce}% _{2m}\left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell% }^{2m}(h^{2}){\cal C}_{2\ell}^{(j)}(2h\cosh z),$
28.23.8 ${\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\operatorname{% ce}_{2m+1}\left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{% 2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
##### 5: 28.1 Special Notation
 $m,n$ integers. … order of the Mathieu function or modified Mathieu function. (When $\nu$ is an integer it is often replaced by $n$.) …
and the modified Mathieu functions
 $\operatorname{Ce}_{\nu}\left(z,q\right)$, $\operatorname{Se}_{\nu}\left(z,q\right)$, $\operatorname{Fe}_{n}\left(z,q\right)$, $\operatorname{Ge}_{n}\left(z,q\right)$, …
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. …
$f_{\mathit{o},n}(h).$
##### 6: 28.35 Tables
###### §28.35 Tables
• Kirkpatrick (1960) contains tables of the modified functions $\operatorname{Ce}_{n}\left(x,q\right)$, $\operatorname{Se}_{n+1}\left(x,q\right)$ for $n=0(1)5$, $q=1(1)20$, $x=0.1(.1)1$; 4D or 5D.

• Zhang and Jin (1996, pp. 521–532) includes the eigenvalues $a_{n}\left(q\right)$, $b_{n+1}\left(q\right)$ for $n=0(1)4$, $q=0(1)50$; $n=0(1)20$ ($a$’s) or 19 ($b$’s), $q=1,3,5,10,15,25,50(50)200$. Fourier coefficients for $\operatorname{ce}_{n}\left(x,10\right)$, $\operatorname{se}_{n+1}\left(x,10\right)$, $n=0(1)7$. Mathieu functions $\operatorname{ce}_{n}\left(x,10\right)$, $\operatorname{se}_{n+1}\left(x,10\right)$, and their first $x$-derivatives for $n=0(1)4$, $x=0(5^{\circ})90^{\circ}$. Modified Mathieu functions ${\operatorname{Mc}^{(j)}_{n}}\left(x,\sqrt{10}\right)$, ${\operatorname{Ms}^{(j)}_{n+1}}\left(x,\sqrt{10}\right)$, and their first $x$-derivatives for $n=0(1)4$, $j=1,2$, $x=0(.2)4$. Precision is mostly 9S.

• 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.

##### 8: 28.33 Physical Applications
###### §28.33 Physical Applications
28.33.2 $V_{n}(\xi,\eta)=\left(c_{n}{\operatorname{M}^{(1)}_{n}}\left(\xi,\sqrt{q}% \right)+d_{n}{\operatorname{M}^{(2)}_{n}}\left(\xi,\sqrt{q}\right)\right)% \operatorname{me}_{n}\left(\eta,q\right),$
28.33.3 ${\operatorname{M}^{(1)}_{n}}\left(\xi_{0},\sqrt{q}\right){\operatorname{M}^{(2% )}_{n}}\left(\xi_{1},\sqrt{q}\right)-{\operatorname{M}^{(1)}_{n}}\left(\xi_{1}% ,\sqrt{q}\right){\operatorname{M}^{(2)}_{n}}\left(\xi_{0},\sqrt{q}\right)=0.$
• Torres-Vega et al. (1998) for Mathieu functions in phase space.

• ##### 9: 28.28 Integrals, Integral Representations, and Integral Equations
###### §28.28(i) Equations with Elementary Kernels
28.28.11 $\int_{0}^{\infty}e^{2\mathrm{i}h\cosh z\cosh t}\operatorname{Ce}_{\nu}\left(t,% h^{2}\right)\,\mathrm{d}t=\tfrac{1}{2}\pi\mathrm{i}e^{\mathrm{i}\nu\pi}% \operatorname{ce}_{\nu}\left(0,h^{2}\right){\operatorname{M}^{(3)}_{\nu}}\left% (z,h\right),$
28.28.15 $\int_{0}^{\infty}\cos\left(2h\cos y\cosh t\right)\operatorname{Ce}_{2n}\left(t% ,h^{2}\right)\,\mathrm{d}t=(-1)^{n+1}\tfrac{1}{2}\pi{\operatorname{Mc}^{(2)}_{% 2n}}\left(0,h\right)\operatorname{ce}_{2n}\left(y,h^{2}\right),$