DLMF:28.23.E4 (Q8388)

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DLMF:28.23.E4
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    Statements

    Defining formula
    me ν ⁡ ( 1 2 ⁢ π , h 2 ) ⁢ M ν ( j ) ⁡ ( z , h ) = e i ⁢ ν ⁢ π / 2 ⁢ ∑ n = - ∞ ∞ c 2 ⁢ n ν ⁢ ( h 2 ) ⁢ 𝒞 ν + 2 ⁢ n ( j ) ⁢ ( 2 ⁢ h ⁢ sinh ⁡ z ) , Mathieu-me 𝜈 1 2 𝜋 superscript ℎ 2 modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ superscript 𝑒 imaginary-unit 𝜈 𝜋 2 superscript subscript 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)=e^{\mathrm{i}\nu\ifrac{\pi}{2}}\sum_% {n=-\infty}^{\infty}c_{2n}^{\nu}(h^{2}){\cal C}_{\nu+2n}^{(j)}(2h\sinh z),}}
    0 references
    DLMF id
    DLMF:28.23.E4
    0 references
    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.m2abdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aadec
    0 references
    Q12301
    Defining formula
    M ν ( j ) ⁡ ( z , h ) modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{M}^{(\NVar{j})}_{\NVar{\nu}}}\left(\NVar{% z},\NVar{h}\right)}}
    xml-id
    C28.S20.SS3.p1.m4abdec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1bdec
    0 references
    Q12279
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1bdec
    0 references
     
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