DLMF:28.23.E5 (Q8389)

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DLMF:28.23.E5
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    Statements

    Defining formula
    me ν ′ ⁡ ( 1 2 ⁢ π , h 2 ) ⁢ M ν ( j ) ⁡ ( z , h ) = i ⁢ e i ⁢ ν ⁢ π / 2 ⁢ coth ⁡ z ⁢ ∑ n = - ∞ ∞ ( ν + 2 ⁢ n ) ⁢ c 2 ⁢ n ν ⁢ ( h 2 ) ⁢ 𝒞 ν + 2 ⁢ n ( j ) ⁢ ( 2 ⁢ h ⁢ sinh ⁡ z ) , diffop Mathieu-me 𝜈 1 1 2 𝜋 superscript ℎ 2 modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ imaginary-unit superscript 𝑒 imaginary-unit 𝜈 𝜋 2 hyperbolic-cotangent 𝑧 superscript subscript 𝑛 𝜈 2 𝑛 superscript subscript 𝑐 2 𝑛 𝜈 superscript ℎ 2 superscript subscript 𝒞 𝜈 2 𝑛 𝑗 2 ℎ 𝑧 {\displaystyle{\displaystyle\mathrm{me}_{\nu}'\left(\tfrac{1}{2}\pi,h^{2}% \right){\mathrm{M}^{(j)}_{\nu}}\left(z,h\right)=\mathrm{i}e^{\mathrm{i}\nu% \ifrac{\pi}{2}}\coth z\sum_{n=-\infty}^{\infty}(\nu+2n)c_{2n}^{\nu}(h^{2}){% \cal C}_{\nu+2n}^{(j)}(2h\sinh z),}}
    0 references
    DLMF id
    DLMF:28.23.E5
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    Symbols used
    Q12299
    Defining formula
    me n ⁡ ( z , q ) Mathieu-me 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{me}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S12.SS2.p2.m2acdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aadec
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