# §28.20 Definitions and Basic Properties

## §28.20(i) Modified Mathieu’s Equation

When $z$ is replaced by $\pm\mathrm{i}z$, (28.2.1) becomes the modified Mathieu’s equation:

 28.20.1 $w^{\prime\prime}-\left(a-2q\mathop{\cosh\/}\nolimits\!\left(2z\right)\right)w=0,$ Symbols: $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $q=h^{2}$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution A&S Ref: 20.1.2 (in slightly different form) 20.8.6 Referenced by: §28.20(iii), §28.32(i), §28.33(i), §28.33(i), (b), §28.8(iv), §28.8(iv) Permalink: http://dlmf.nist.gov/28.20.E1 Encodings: TeX, pMML, png See also: Annotations for 28.20(i)

with its algebraic form

 28.20.2 ${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$ $\zeta=\mathop{\cosh\/}\nolimits z$.

## §28.20(ii) Solutions $\mathop{\mathrm{Ce}_{\nu}\/}\nolimits$, $\mathop{\mathrm{Se}_{\nu}\/}\nolimits$, $\mathop{\mathrm{Me}_{\nu}\/}\nolimits$, $\mathop{\mathrm{Fe}_{n}\/}\nolimits$, $\mathop{\mathrm{Ge}_{n}\/}\nolimits$

 28.20.3 $\displaystyle\mathop{\mathrm{Ce}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\mathrm{ce}_{\nu}\/}\nolimits\!\left(\pm\mathrm{i}z,q% \right),$ $\nu\neq-1,-2,\dots$, Defines: $\mathop{\mathrm{Ce}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathop{\mathrm{ce}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $q=h^{2}$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E3 Encodings: TeX, pMML, png See also: Annotations for 28.20(ii) 28.20.4 $\displaystyle\mathop{\mathrm{Se}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mp\mathrm{i}\mathop{\mathrm{se}_{\nu}\/}\nolimits\!\left(\pm% \mathrm{i}z,q\right),$ $\nu\neq 0,-1,\dots$, Defines: $\mathop{\mathrm{Se}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathop{\mathrm{se}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $q=h^{2}$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E4 Encodings: TeX, pMML, png See also: Annotations for 28.20(ii) 28.20.5 $\displaystyle\mathop{\mathrm{Me}_{\nu}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\mathrm{me}_{\nu}\/}\nolimits\!\left(-\mathrm{i}z,q% \right),$ Defines: $\mathop{\mathrm{Me}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathop{\mathrm{me}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: Mathieu function, $q=h^{2}$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E5 Encodings: TeX, pMML, png See also: Annotations for 28.20(ii) 28.20.6 $\displaystyle\mathop{\mathrm{Fe}_{n}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mp\mathrm{i}\mathop{\mathrm{fe}_{n}\/}\nolimits\!\left(\pm% \mathrm{i}z,q\right),$ $n=0,1,\dots$, Defines: $\mathop{\mathrm{Fe}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathop{\mathrm{fe}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: second solution, Mathieu’s equation, $q=h^{2}$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E6 Encodings: TeX, pMML, png See also: Annotations for 28.20(ii) 28.20.7 $\displaystyle\mathop{\mathrm{Ge}_{n}\/}\nolimits\!\left(z,q\right)$ $\displaystyle=\mathop{\mathrm{ge}_{n}\/}\nolimits\!\left(\pm\mathrm{i}z,q% \right),$ $n=1,2,\dots$. Defines: $\mathop{\mathrm{Ge}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathop{\mathrm{ge}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: second solution, Mathieu’s equation, $q=h^{2}$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.6.1 (in slightly different form) 20.6.2 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E7 Encodings: TeX, pMML, png See also: Annotations for 28.20(ii)

## §28.20(iii) Solutions $\mathop{{\mathrm{M}^{(j)}_{\nu}}\/}\nolimits$

Assume first that $\nu$ is real, $q$ is positive, and $a=\mathop{\lambda_{\nu}\/}\nolimits\!\left(q\right)$; see §28.12(i). Write

 28.20.8 $h=\sqrt{q}\;(>0).$ Symbols: $q=h^{2}$: parameter and $h$: parameter Permalink: http://dlmf.nist.gov/28.20.E8 Encodings: TeX, pMML, png See also: Annotations for 28.20(iii)

Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to $\zeta^{\ifrac{1}{2}}e^{\pm 2\mathrm{i}h\zeta}$ as $\zeta\to\infty$ in the respective sectors $|\mathop{\mathrm{ph}\/}\nolimits\!\left(\mp\mathrm{i}\zeta\right)|\leq\tfrac{3% }{2}\pi-\delta$, $\delta$ being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions $\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\!\left(z,h\right)$, $\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\!\left(z,h\right)$ such that

 28.20.9 $\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\!\left(z,h\right)=\mathop{{H^{(1)% }_{\nu}}\/}\nolimits\!\left(2h\mathop{\cosh\/}\nolimits z\right)\left(1+% \mathop{O\/}\nolimits\!\left(\mathop{\mathrm{sech}\/}\nolimits z\right)\right),$

as $\Re{z}\to+\infty$ with $-\pi+\delta\leq\Im{z}\leq 2\pi-\delta$, and

 28.20.10 $\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\!\left(z,h\right)=\mathop{{H^{(2)% }_{\nu}}\/}\nolimits\!\left(2h\mathop{\cosh\/}\nolimits z\right)\left(1+% \mathop{O\/}\nolimits\!\left(\mathop{\mathrm{sech}\/}\nolimits z\right)\right),$

as $\Re{z}\to+\infty$ with $-2\pi+\delta\leq\Im{z}\leq\pi-\delta$. See §10.2(ii) for the notation. In addition, there are unique solutions $\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!\left(z,h\right)$, $\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}\nolimits\!\left(z,h\right)$ that are real when $z$ is real and have the properties

 28.20.11 $\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!\left(z,h\right)=\mathop{J_{\nu}% \/}\nolimits\!\left(2h\mathop{\cosh\/}\nolimits z\right)+e^{|\Im{(2h\mathop{% \cosh\/}\nolimits z)}|}\mathop{O\/}\nolimits\!\left(\left(\mathop{\mathrm{sech% }\/}\nolimits z\right)^{3/2}\right),$
 28.20.12 $\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}\nolimits\!\left(z,h\right)=\mathop{Y_{\nu}% \/}\nolimits\!\left(2h\mathop{\cosh\/}\nolimits z\right)+e^{|\Im{(2h\mathop{% \cosh\/}\nolimits z)}|}\mathop{O\/}\nolimits\!\left((\mathop{\mathrm{sech}\/}% \nolimits z)^{3/2}\right),$

as $\Re{z}\to+\infty$ with $|\Im{z}|\leq\pi-\delta$.

For other values of $z$, $h,$ and $\nu$ the functions $\mathop{{\mathrm{M}^{(j)}_{\nu}}\/}\nolimits\!\left(z,h\right)$, $j=1,2,3,4,$ are determined by analytic continuation. Furthermore,

 28.20.13 $\displaystyle\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!\left(z,h\right)+% \mathrm{i}\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}\nolimits\!\left(z,h\right),$ Symbols: $\mathop{{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter A&S Ref: 20.6.16 (only for integer $\nu$) Referenced by: §28.22(ii) Permalink: http://dlmf.nist.gov/28.20.E13 Encodings: TeX, pMML, png See also: Annotations for 28.20(iii) 28.20.14 $\displaystyle\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!\left(z,h\right)-% \mathrm{i}\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}\nolimits\!\left(z,h\right).$

## §28.20(iv) Radial Mathieu Functions $\mathop{{\mathrm{Mc}^{(j)}_{n}}\/}\nolimits$, $\mathop{{\mathrm{Ms}^{(j)}_{n}}\/}\nolimits$

For $j=1,2,3,4$,

 28.20.15 $\displaystyle\mathop{{\mathrm{Mc}^{(j)}_{n}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=\mathop{{\mathrm{M}^{(j)}_{n}}\/}\nolimits\!\left(z,h\right),$ $n=0,1,\dots$, Defines: $\mathop{{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function Symbols: $\mathop{{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable Referenced by: §28.20(vii) Permalink: http://dlmf.nist.gov/28.20.E15 Encodings: TeX, pMML, png See also: Annotations for 28.20(iv) 28.20.16 $\displaystyle\mathop{{\mathrm{Ms}^{(j)}_{n}}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=(-1)^{n}\mathop{{\mathrm{M}^{(j)}_{-n}}\/}\nolimits\!\left(z,h% \right),$ $n=1,2,\dots$. Defines: $\mathop{{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function Symbols: $\mathop{{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable Referenced by: §28.20(vii) Permalink: http://dlmf.nist.gov/28.20.E16 Encodings: TeX, pMML, png See also: Annotations for 28.20(iv)

## §28.20(v) Solutions $\mathop{\mathrm{Ie}_{n}\/}\nolimits$, $\mathop{\mathrm{Io}_{n}\/}\nolimits$, $\mathop{\mathrm{Ke}_{n}\/}\nolimits$, $\mathop{\mathrm{Ko}_{n}\/}\nolimits$

 28.20.17 $\displaystyle\mathop{\mathrm{Ie}_{n}\/}\nolimits\!\left(z,h\right)$ $\displaystyle={\mathrm{i}^{-n}}\mathop{{\mathrm{Mc}^{(1)}_{n}}\/}\nolimits\!% \left(z,\mathrm{i}h\right),$ Defines: $\mathop{\mathrm{Ie}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function Symbols: $\mathop{{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.8.10 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E17 Encodings: TeX, pMML, png See also: Annotations for 28.20(v) 28.20.18 $\displaystyle\mathop{\mathrm{Io}_{n}\/}\nolimits\!\left(z,h\right)$ $\displaystyle={\mathrm{i}^{-n}}\mathop{{\mathrm{Ms}^{(1)}_{n}}\/}\nolimits\!% \left(z,\mathrm{i}h\right),$ Defines: $\mathop{\mathrm{Io}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function Symbols: $\mathop{{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.8.10 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E18 Encodings: TeX, pMML, png See also: Annotations for 28.20(v)
 28.20.19 $\displaystyle\mathop{\mathrm{Ke}_{2m}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=(-1)^{m}\tfrac{1}{2}\pi\mathrm{i}\mathop{{\mathrm{Mc}^{(3)}_{2m}% }\/}\nolimits\!\left(z,\mathrm{i}h\right),$ $\displaystyle\mathop{\mathrm{Ke}_{2m+1}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=(-1)^{m+1}\tfrac{1}{2}\pi\mathop{{\mathrm{Mc}^{(3)}_{2m+1}}\/}% \nolimits\!\left(z,\mathrm{i}h\right),$ Defines: $\mathop{\mathrm{Ke}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $m$: integer, $h$: parameter, $n$: integer and $z$: complex variable A&S Ref: 20.8.11 (in slightly different form) Permalink: http://dlmf.nist.gov/28.20.E19 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 28.20(v)
 28.20.20 $\displaystyle\mathop{\mathrm{Ko}_{2m}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=(-1)^{m}\tfrac{1}{2}\pi\mathrm{i}\mathop{{\mathrm{Ms}^{(3)}_{2m}% }\/}\nolimits\!\left(z,\mathrm{i}h\right),$ $\displaystyle\mathop{\mathrm{Ko}_{2m+1}\/}\nolimits\!\left(z,h\right)$ $\displaystyle=(-1)^{m+1}\tfrac{1}{2}\pi\mathop{{\mathrm{Ms}^{(3)}_{2m+1}}\/}% \nolimits\!\left(z,\mathrm{i}h\right).$ Defines: $\mathop{\mathrm{Ko}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\/}\nolimits\!\left(\NVar{z},% \NVar{h}\right)$: radial Mathieu function, $m$: integer, $h$: parameter, $n$: integer and $z$: complex variable Permalink: http://dlmf.nist.gov/28.20.E20 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 28.20(v)

## §28.20(vi) Wronskians

 28.20.21 $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathrm{M}^{(1)}_{% \nu}}\/}\nolimits,\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}\nolimits\right\}$ $\displaystyle=-\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathrm{M}^{(2)}% _{\nu}}\/}\nolimits,\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\right\}=-% \mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}% \nolimits,\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\right\}=\ifrac{2}{\pi},$ $\displaystyle\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathrm{M}^{(1)}_{% \nu}}\/}\nolimits,\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\right\}$ $\displaystyle=-\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathrm{M}^{(1)}% _{\nu}}\/}\nolimits,\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\right\}=-% \tfrac{1}{2}\mathop{\mathscr{W}\/}\nolimits\left\{\mathop{{\mathrm{M}^{(3)}_{% \nu}}\/}\nolimits,\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\right\}=\ifrac{% 2\mathrm{i}}{\pi}.$

## §28.20(vii) Shift of Variable

 28.20.22 $\mathop{{\mathrm{M}^{(j)}_{\nu}}\/}\nolimits\!\left(z\pm\tfrac{1}{2}\pi\mathrm% {i},h\right)=\mathop{{\mathrm{M}^{(j)}_{\nu}}\/}\nolimits\!\left(z,\pm\mathrm{% i}h\right),$ $\nu\notin\mathbb{Z}$.

For $n=0,1,2,\dots$,

 28.20.23 $\displaystyle\mathop{{\mathrm{Mc}^{(j)}_{2n}}\/}\nolimits\!\left(z\pm\tfrac{1}% {2}\pi\mathrm{i},h\right)$ $\displaystyle=\mathop{{\mathrm{Mc}^{(j)}_{2n}}\/}\nolimits\!\left(z,\pm\mathrm% {i}h\right),$ $\displaystyle\mathop{{\mathrm{Ms}^{(j)}_{2n+1}}\/}\nolimits\!\left(z\pm\tfrac{% 1}{2}\pi\mathrm{i},h\right)$ $\displaystyle=\mathop{{\mathrm{Mc}^{(j)}_{2n+1}}\/}\nolimits\!\left(z,\pm% \mathrm{i}h\right),$
 28.20.24 $\displaystyle\mathop{{\mathrm{Mc}^{(j)}_{2n+1}}\/}\nolimits\!\left(z\pm\tfrac{% 1}{2}\pi\mathrm{i},h\right)$ $\displaystyle=\mathop{{\mathrm{Ms}^{(j)}_{2n+1}}\/}\nolimits\!\left(z,\pm% \mathrm{i}h\right),$ $\displaystyle\mathop{{\mathrm{Ms}^{(j)}_{2n+2}}\/}\nolimits\!\left(z\pm\tfrac{% 1}{2}\pi\mathrm{i},h\right)$ $\displaystyle=\mathop{{\mathrm{Ms}^{(j)}_{2n+2}}\/}\nolimits\!\left(z,\pm% \mathrm{i}h\right).$

For $s\in\mathbb{Z}$,

 28.20.25 $\displaystyle\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}\nolimits\!\left(z+s\pi\mathrm% {i},h\right)$ $\displaystyle=e^{\mathrm{i}s\pi\nu}\mathop{{\mathrm{M}^{(1)}_{\nu}}\/}% \nolimits\!\left(z,h\right),$ $\displaystyle\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}\nolimits\!\left(z+s\pi\mathrm% {i},h\right)$ $\displaystyle=e^{-\mathrm{i}s\pi\nu}\mathop{{\mathrm{M}^{(2)}_{\nu}}\/}% \nolimits\!\left(z,h\right)+2\mathrm{i}\mathop{\cot\/}\nolimits\!\left(\pi\nu% \right)\mathop{\sin\/}\nolimits\!\left(s\pi\nu\right)\mathop{{\mathrm{M}^{(1)}% _{\nu}}\/}\nolimits\!\left(z,h\right),$ $\displaystyle\mathop{{\mathrm{M}^{(3)}_{\nu}}\/}\nolimits\!\left(z+s\pi\mathrm% {i},h\right)$ $\displaystyle=-\dfrac{\mathop{\sin\/}\nolimits\!\left({(s-1)\pi\nu}\right)}{% \mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}\mathop{{\mathrm{M}^{(3)}_{\nu}}% \/}\nolimits\!\left(z,h\right)-e^{-\mathrm{i}\pi\nu}\frac{\mathop{\sin\/}% \nolimits\!\left(s\pi\nu\right)}{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)% }\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\!\left(z,h\right),$ $\displaystyle\mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\!\left(z+s\pi\mathrm% {i},h\right)$ $\displaystyle=e^{\mathrm{i}\pi\nu}\dfrac{\mathop{\sin\/}\nolimits\!\left(s\pi% \nu\right)}{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}\mathop{{\mathrm{M}^% {(3)}_{\nu}}\/}\nolimits\!\left(z,h\right)+\frac{\mathop{\sin\/}\nolimits\!% \left((s+1)\pi\nu\right)}{\mathop{\sin\/}\nolimits\!\left(\pi\nu\right)}% \mathop{{\mathrm{M}^{(4)}_{\nu}}\/}\nolimits\!\left(z,h\right).$

When $\nu$ is an integer the right-hand sides of (28.20.25) are replaced by the their limiting values. And for the corresponding identities for the radial functions use (28.20.15) and (28.20.16).