28 Mathieu Functions and Hill’s Equation Modified Mathieu Functions 28.24 Expansions in Series of Cross-Products of Bessel Functions or Modified Bessel Functions 28.26 Asymptotic Approximations for Large

§28.25 Asymptotic Expansions for Large $\realpart{z}$

Keywords:: modified Mathieu functions
Referenced by:: §28.34(iv)
Permalink:: http://dlmf.nist.gov/28.25

For fixed $h(\neq 0)$ and fixed $\nu$ ,

28.25.1 $\mathop{{\mathrm{M}^{{(3,4)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)\sim\frac{% e^{{\pm i\left(2h\mathop{\cosh\/}\nolimits z-\left(\frac{1}{2}\nu+\frac{1}{4}% \right)\pi\right)}}}{\left(\pi h(\mathop{\cosh\/}\nolimits z+1)\right)^{{\frac% {1}{2}}}}\*\sum_{{m=0}}^{{\infty}}\dfrac{D^{{\pm}}_{m}}{\left(\mp 4ih(\mathop{% \cosh\/}\nolimits z+1)\right)^{m}},$

Symbols:: : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{{\mathrm{M}^{{(j)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ : modified Mathieu function, $\sim$ : asymptotic equality, : integer, : parameter, : complex variable, $\nu$ : complex parameter and $D_{m}^{{\pm}}$ : coefficients
A&S Ref:: 20.9.1 (for 3 and for integer $\nu$ ) 20.9.3 (for 4 and for integer $\nu$ )
Referenced by:: §28.25
Permalink:: http://dlmf.nist.gov/28.25.E1
Encodings:: TeX, pMML, png

where the coefficients are given by

28.25.2

$D_{{-1}}^{{\pm}}=0,$

$D_{{0}}^{{\pm}}=1,$

Symbols:: : integer and $D_{m}^{{\pm}}$ : coefficients
Permalink:: http://dlmf.nist.gov/28.25.E2
Encodings:: TeX, TeX, pMML, pMML, png, png

and

28.25.3 $(m+1)D^{{\pm}}_{{m+1}}+{\left((m+\tfrac{1}{2})^{2}\pm(m+\tfrac{1}{4})8ih+2h^{2% }-a\right)D^{{\pm}}_{m}}\pm(m-\tfrac{1}{2})\left(8ihm\right)D_{{m-1}}^{{\pm}}=0,$ $m\geq 0$ .

Symbols:: : integer, : parameter, : parameter and $D_{m}^{{\pm}}$ : coefficients
A&S Ref:: 20.9.2 (for ) 20.9.4 (for )
Permalink:: http://dlmf.nist.gov/28.25.E3
Encodings:: TeX, pMML, png

The upper signs correspond to $\mathop{{\mathrm{M}^{{(3)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ and the lower signs to $\mathop{{\mathrm{M}^{{(4)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ . The expansion (28.25.1) is valid for $\mathop{{\mathrm{M}^{{(3)}}_{{\nu}}}\/}\nolimits\!\left(z,h\right)$ when