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## 1—10 of 14 matching pages

##### 1: 28.20 Definitions and Basic Properties
###### §28.20(iv) RadialMathieuFunctions${\mathrm{Mc}^{(j)}_{n}}$, ${\mathrm{Ms}^{(j)}_{n}}$
28.20.15 ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)={\mathrm{M}^{(j)}_{n}}\left(z,h\right),$ $n=0,1,\dots$,
28.20.16 ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)=(-1)^{n}{\mathrm{M}^{(j)}_{-n}}\left(z% ,h\right),$ $n=1,2,\dots$.
28.20.17 $\mathrm{Ie}_{n}\left(z,h\right)={\mathrm{i}}^{-n}{\mathrm{Mc}^{(1)}_{n}}\left(% z,\mathrm{i}h\right),$
##### 2: 28.21 Graphics Figure 28.21.6: Ms 1 ( 2 ) ⁡ ( x , h ) for 0.2 ≤ h ≤ 2 , 0 ≤ x ≤ 2 . Magnify 3D Help
##### 3: 28.22 Connection Formulas
28.22.1 ${\mathrm{Mc}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\dfrac{1}{g_{% \mathit{e},m}(h)\mathrm{ce}_{m}\left(0,h^{2}\right)}\mathrm{Ce}_{m}\left(z,h^{% 2}\right),$
28.22.2 ${\mathrm{Ms}^{(1)}_{m}}\left(z,h\right)=\sqrt{\dfrac{2}{\pi}}\frac{1}{g_{% \mathit{o},m}(h)\mathrm{se}_{m}'\left(0,h^{2}\right)}\mathrm{Se}_{m}\left(z,h^% {2}\right),$
The joining factors in the above formulas are given by …
28.22.9 $f_{\mathit{e},m}(h)=-\sqrt{\ifrac{\pi}{2}}g_{\mathit{e},m}(h){\mathrm{Mc}^{(2)% }_{m}}\left(0,h\right),$
$\mathrm{ge}_{m}\left(0,h^{2}\right)=\tfrac{1}{2}\pi S_{m}(h^{2})\left(g_{% \mathit{o},m}(h)\right)^{2}\mathrm{se}_{m}'\left(0,h^{2}\right).$
##### 4: 28.23 Expansions in Series of Bessel Functions
###### §28.23 Expansions in Series of Bessel Functions
28.23.6 ${\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\mathrm{ce}_{2m}\left(0% ,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell}^{2m}(h^{2}% ){\cal C}_{2\ell}^{(j)}(2h\cosh z),$
28.23.7 ${\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\left(\mathrm{ce}_{2m}\left(% \tfrac{1}{2}\pi,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}A_{2\ell}^{2m}(h% ^{2}){\cal C}_{2\ell}^{(j)}(2h\sinh z),$
28.23.8 ${\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\left(\mathrm{ce}_{2m+1}% \left(0,h^{2}\right)\right)^{-1}\sum_{\ell=0}^{\infty}(-1)^{\ell}A_{2\ell+1}^{% 2m+1}(h^{2}){\cal C}_{2\ell+1}^{(j)}(2h\cosh z),$
##### 5: 28.28 Integrals, Integral Representations, and Integral Equations
###### §28.28(i) Equations with Elementary Kernels
28.28.23 $\dfrac{2}{\pi}\int_{0}^{\pi}\mathcal{C}^{(j)}_{2\ell+2}(2hR)\sin\left((2\ell+2% )\phi\right)\mathrm{se}_{2m+2}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{\ell+m}B^{% 2m+2}_{2\ell+2}(h^{2}){\mathrm{Ms}^{(j)}_{2m+2}}\left(z,h\right).$
###### §28.28(iv) Integrals of Products of MathieuFunctions of Integer Order
28.28.49 $\widehat{\alpha}_{n,m}^{(c)}=\frac{1}{2\pi}\int_{0}^{2\pi}\cos t\mathrm{ce}_{n% }\left(t,h^{2}\right)\mathrm{ce}_{m}\left(t,h^{2}\right)\mathrm{d}t=(-1)^{p+1}% \dfrac{2}{\mathrm{i}\pi}\dfrac{\mathrm{ce}_{n}\left(0,h^{2}\right)\mathrm{ce}_% {m}\left(0,h^{2}\right)}{h\mathrm{Dc}_{0}\left(n,m,0\right)}.$
##### 6: 28.1 Special Notation
 $\mathrm{Ce}_{\nu}\left(z,q\right)$, $\mathrm{Se}_{\nu}\left(z,q\right)$, $\mathrm{Fe}_{n}\left(z,q\right)$, $\mathrm{Ge}_{n}\left(z,q\right)$, …
The functions ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$ are also known as the radial Mathieu functions. …
$f_{\mathit{o},n}(h).$
The radial functions ${\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$ are denoted by ${\mathrm{Mc}^{(j)}_{n}}\left(z,q\right)$ and ${\mathrm{Ms}^{(j)}_{n}}\left(z,q\right)$, 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 $\varepsilon_{s}{\mathrm{Mc}^{(j)}_{2m}}\left(z,h\right)=(-1)^{m}\sum_{\ell=0}^% {\infty}(-1)^{\ell}\frac{A_{2\ell}^{2m}(h^{2})}{A_{2s}^{2m}(h^{2})}\left(J_{% \ell-s}\left(he^{-z}\right){\cal C}_{\ell+s}^{(j)}(he^{z})+J_{\ell+s}\left(he^% {-z}\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
28.24.3 ${\mathrm{Mc}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^% {\ell}\frac{A_{2\ell+1}^{2m+1}(h^{2})}{A_{2s+1}^{2m+1}(h^{2})}\left(J_{\ell-s}% \left(he^{-z}\right){\cal C}_{\ell+s+1}^{(j)}(he^{z})+J_{\ell+s+1}\left(he^{-z% }\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
28.24.4 ${\mathrm{Ms}^{(j)}_{2m+1}}\left(z,h\right)=(-1)^{m}\sum_{\ell=0}^{\infty}(-1)^% {\ell}\frac{B_{2\ell+1}^{2m+1}(h^{2})}{B_{2s+1}^{2m+1}(h^{2})}\left(J_{\ell-s}% \left(he^{-z}\right){\cal C}_{\ell+s+1}^{(j)}(he^{z})-J_{\ell+s+1}\left(he^{-z% }\right){\cal C}_{\ell-s}^{(j)}(he^{z})\right),$
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 ${\mathrm{Mc}^{(3)}_{m}}\left(z,h\right)=\dfrac{e^{\mathrm{i}\phi}}{(\pi h\cosh z% )^{\ifrac{1}{2}}}\*\left(\mathrm{Fc}_{m}\left(z,h\right)-\mathrm{i}\mathrm{Gc}% _{m}\left(z,h\right)\right),$
28.26.2 $\mathrm{i}{\mathrm{Ms}^{(3)}_{m+1}}\left(z,h\right)=\dfrac{e^{\mathrm{i}\phi}}% {(\pi h\cosh z)^{\ifrac{1}{2}}}\*{\left(\mathrm{Fs}_{m}\left(z,h\right)-% \mathrm{i}\mathrm{Gs}_{m}\left(z,h\right)\right)},$
##### 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.