§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 $w^{\prime\prime}-\left(a-2q\cosh\left(2z\right)\right)w=0,$ ⓘ Symbols: $\cosh\NVar{z}$: 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), 28.20 and 28

with its algebraic form

 28.20.2 ${(\zeta^{2}-1)w^{\prime\prime}+\zeta w^{\prime}+\left(4q\zeta^{2}-2q-a\right)w% =0},$ $\zeta=\cosh z$. ⓘ Symbols: $\cosh\NVar{z}$: hyperbolic cosine function, $q=h^{2}$: parameter, $z$: complex variable, $a$: parameter and $w(z)$: Mathieu’s equation solution Referenced by: §28.20(iii) Permalink: http://dlmf.nist.gov/28.20.E2 Encodings: TeX, pMML, png See also: Annotations for 28.20(i), 28.20 and 28

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

 28.20.3 $\displaystyle\mathrm{Ce}_{\nu}\left(z,q\right)$ $\displaystyle=\mathrm{ce}_{\nu}\left(\pm\mathrm{i}z,q\right),$ $\nu\neq-1,-2,\dots$, ⓘ Defines: $\mathrm{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: 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 and 28 28.20.4 $\displaystyle\mathrm{Se}_{\nu}\left(z,q\right)$ $\displaystyle=\mp\mathrm{i}\mathrm{se}_{\nu}\left(\pm\mathrm{i}z,q\right),$ $\nu\neq 0,-1,\dots$, ⓘ Defines: $\mathrm{Se}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathrm{se}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: 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 and 28 28.20.5 $\displaystyle\mathrm{Me}_{\nu}\left(z,q\right)$ $\displaystyle=\mathrm{me}_{\nu}\left(-\mathrm{i}z,q\right),$ ⓘ Defines: $\mathrm{Me}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: 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 and 28 28.20.6 $\displaystyle\mathrm{Fe}_{n}\left(z,q\right)$ $\displaystyle=\mp\mathrm{i}\mathrm{fe}_{n}\left(\pm\mathrm{i}z,q\right),$ $n=0,1,\dots$, ⓘ Defines: $\mathrm{Fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathrm{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: 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 and 28 28.20.7 $\displaystyle\mathrm{Ge}_{n}\left(z,q\right)$ $\displaystyle=\mathrm{ge}_{n}\left(\pm\mathrm{i}z,q\right),$ $n=1,2,\dots$. ⓘ Defines: $\mathrm{Ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: modified Mathieu function Symbols: $\mathrm{ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}\right)$: 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 and 28

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

Assume first that $\nu$ is real, $q$ is positive, and $a=\lambda_{\nu}\left(q\right)$; 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), 28.20 and 28

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 $|\operatorname{ph}\left(\mp\mathrm{i}\zeta\right)|\leq\tfrac{3}{2}\pi-\delta$, $\delta$ being an arbitrary small positive constant. It follows that (28.20.1) has independent and unique solutions ${\mathrm{M}^{(3)}_{\nu}}\left(z,h\right)$, ${\mathrm{M}^{(4)}_{\nu}}\left(z,h\right)$ such that

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

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

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

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 ${\mathrm{M}^{(1)}_{\nu}}\left(z,h\right)$, ${\mathrm{M}^{(2)}_{\nu}}\left(z,h\right)$ that are real when $z$ is real and have the properties

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

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

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

 28.20.13 $\displaystyle{\mathrm{M}^{(3)}_{\nu}}\left(z,h\right)$ $\displaystyle={\mathrm{M}^{(1)}_{\nu}}\left(z,h\right)+\mathrm{i}{\mathrm{M}^{% (2)}_{\nu}}\left(z,h\right),$ ⓘ Symbols: ${\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28 28.20.14 $\displaystyle{\mathrm{M}^{(4)}_{\nu}}\left(z,h\right)$ $\displaystyle={\mathrm{M}^{(1)}_{\nu}}\left(z,h\right)-\mathrm{i}{\mathrm{M}^{% (2)}_{\nu}}\left(z,h\right).$ ⓘ Symbols: ${\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter Referenced by: §28.22(ii) Permalink: http://dlmf.nist.gov/28.20.E14 Encodings: TeX, pMML, png See also: Annotations for 28.20(iii), 28.20 and 28

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

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

 28.20.15 $\displaystyle{\mathrm{Mc}^{(j)}_{n}}\left(z,h\right)$ $\displaystyle={\mathrm{M}^{(j)}_{n}}\left(z,h\right),$ $n=0,1,\dots$, ⓘ Defines: ${\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: radial Mathieu function Symbols: ${\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28 28.20.16 $\displaystyle{\mathrm{Ms}^{(j)}_{n}}\left(z,h\right)$ $\displaystyle=(-1)^{n}{\mathrm{M}^{(j)}_{-n}}\left(z,h\right),$ $n=1,2,\dots$. ⓘ Defines: ${\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: radial Mathieu function Symbols: ${\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28

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

 28.20.17 $\displaystyle\mathrm{Ie}_{n}\left(z,h\right)$ $\displaystyle={\mathrm{i}^{-n}}{\mathrm{Mc}^{(1)}_{n}}\left(z,\mathrm{i}h% \right),$ ⓘ Defines: $\mathrm{Ie}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function Symbols: ${\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28 28.20.18 $\displaystyle\mathrm{Io}_{n}\left(z,h\right)$ $\displaystyle={\mathrm{i}^{-n}}{\mathrm{Ms}^{(1)}_{n}}\left(z,\mathrm{i}h% \right),$ ⓘ Defines: $\mathrm{Io}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function Symbols: ${\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28
 28.20.19 $\displaystyle\mathrm{Ke}_{2m}\left(z,h\right)$ $\displaystyle=(-1)^{m}\tfrac{1}{2}\pi\mathrm{i}{\mathrm{Mc}^{(3)}_{2m}}\left(z% ,\mathrm{i}h\right),$ $\displaystyle\mathrm{Ke}_{2m+1}\left(z,h\right)$ $\displaystyle=(-1)^{m+1}\tfrac{1}{2}\pi{\mathrm{Mc}^{(3)}_{2m+1}}\left(z,% \mathrm{i}h\right),$ ⓘ Defines: $\mathrm{Ke}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28
 28.20.20 $\displaystyle\mathrm{Ko}_{2m}\left(z,h\right)$ $\displaystyle=(-1)^{m}\tfrac{1}{2}\pi\mathrm{i}{\mathrm{Ms}^{(3)}_{2m}}\left(z% ,\mathrm{i}h\right),$ $\displaystyle\mathrm{Ko}_{2m+1}\left(z,h\right)$ $\displaystyle=(-1)^{m+1}\tfrac{1}{2}\pi{\mathrm{Ms}^{(3)}_{2m+1}}\left(z,% \mathrm{i}h\right).$ ⓘ Defines: $\mathrm{Ko}_{\NVar{n}}\left(\NVar{z},\NVar{h}\right)$: modified Mathieu function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z},\NVar{h}\right)$: 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 and 28

§28.20(vi) Wronskians

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

§28.20(vii) Shift of Variable

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

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

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

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

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

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).