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## 1—10 of 15 matching pages

##### 1: 28.20 Definitions and Basic Properties
###### §28.20(iv) RadialMathieuFunctions${\operatorname{Mc}^{(j)}_{n}}$, ${\operatorname{Ms}^{(j)}_{n}}$
28.20.15 ${\operatorname{Mc}^{(j)}_{n}}\left(z,h\right)={\operatorname{M}^{(j)}_{n}}% \left(z,h\right),$ $n=0,1,\dots$,
28.20.16 ${\operatorname{Ms}^{(j)}_{n}}\left(z,h\right)=(-1)^{n}{\operatorname{M}^{(j)}_% {-n}}\left(z,h\right),$ $n=1,2,\dots$.
28.20.17 $\operatorname{Ie}_{n}\left(z,h\right)={\mathrm{i}}^{-n}{\operatorname{Mc}^{(1)% }_{n}}\left(z,\mathrm{i}h\right),$
##### 3: 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 ${\operatorname{Ms}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\frac{1}{g_% {\mathit{o},m}(h)\operatorname{se}_{m}'\left(0,h^{2}\right)}\operatorname{Se}_% {m}\left(z,h^{2}\right),$
The joining factors in the above formulas are given by …
28.22.9 $f_{\mathit{e},m}(h)=-\sqrt{\ifrac{\pi}{2}}g_{\mathit{e},m}(h){\operatorname{Mc% }^{(2)}_{m}}\left(0,h\right),$
$\operatorname{ge}_{m}\left(0,h^{2}\right)=\tfrac{1}{2}\pi S_{m}(h^{2})\left(g_% {\mathit{o},m}(h)\right)^{2}\operatorname{se}_{m}'\left(0,h^{2}\right).$
##### 4: 28.23 Expansions in Series of Bessel Functions
###### §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 ${\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\operatorname{% ce}_{2m+1}\left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{% 2\ell+1}^{2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
##### 5: 28.28 Integrals, Integral Representations, and Integral Equations
###### §28.28(i) Equations with Elementary Kernels
28.28.23 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\sin\left((2\ell+2% )\phi\right)\operatorname{se}_{2m+2}\left(t,h^{2}\right)\,\mathrm{d}t=(-1)^{% \ell+m}B^{2m+2}_{2\ell+2}(h^{2}){\operatorname{Ms}^{(j)}_{2m+2}}\left(z,h% \right).$
###### §28.28(iv) Integrals of Products of MathieuFunctions of Integer Order
28.28.49 $\widehat{\alpha}_{n,m}^{(c)}=\frac{1}{2\pi}\int_{0}^{2\pi}\cos t\operatorname{% ce}_{n}\left(t,h^{2}\right)\operatorname{ce}_{m}\left(t,h^{2}\right)\,\mathrm{% d}t=(-1)^{p+1}\dfrac{2}{\mathrm{i}\pi}\dfrac{\operatorname{ce}_{n}\left(0,h^{2% }\right)\operatorname{ce}_{m}\left(0,h^{2}\right)}{h\operatorname{Dc}_{0}\left% (n,m,0\right)}.$
##### 6: 28.1 Special Notation
 $\operatorname{Ce}_{\nu}\left(z,q\right)$, $\operatorname{Se}_{\nu}\left(z,q\right)$, $\operatorname{Fe}_{n}\left(z,q\right)$, $\operatorname{Ge}_{n}\left(z,q\right)$, …
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. …
$f_{\mathit{o},n}(h).$
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.
##### 7: 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
###### §28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions
28.24.2 $\varepsilon_{s}{\operatorname{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\sum_{% \ell=0}^{\infty}(-1)^{\ell}\frac{A_{2\ell}^{2m}(h^{2})}{A_{2s}^{2m}(h^{2})}% \left(J_{\ell-s}\left(he^{-z}\right){\cal C}_{\ell+s}^{(j)}(he^{z})+J_{\ell+s}% \left(he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
28.24.3 ${\operatorname{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\sum_{\ell=0}^{\infty% }(-1)^{\ell}\frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(J_{% \ell-s}\left(he^{-z}\right){\cal C}_{\ell+s+1}^{(j)}(he^{z})+J_{\ell+s+1}\left% (he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
28.24.4 ${\operatorname{Ms}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\sum_{\ell=0}^{\infty% }(-1)^{\ell}\frac{B_{2\ell+1}^{2m+1}(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(J_{% \ell-s}\left(he^{-z}\right){\cal C}_{\ell+s+1}^{(j)}(he^{z})-J_{\ell+s+1}\left% (he^{-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
For further power series of Mathieu radial functions of integer order for small parameters and improved convergence rate see Larsen et al. (2009).
##### 8: 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 $\mathrm{i}{\operatorname{Ms}^{(3)}_{m+1}}\left(z,h\right)=\dfrac{e^{\mathrm{i}% \phi}}{(\pi h\cosh z)^{\ifrac{1}{2}}}\*{\left(\mathrm{Fs}_{m}\left(z,h\right)-% \mathrm{i}\mathrm{Gs}_{m}\left(z,h\right)\right)},$
##### 9: Bibliography B
• G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..
• G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..
• ##### 10: Bibliography L
• T. M. Larsen, D. Erricolo, and P. L. E. Uslenghi (2009) New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.