DLMF:28.28.E2 (Q8423)

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DLMF:28.28.E2
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    Statements

    Defining formula
    1 2 ⁢ π ⁢ ∫ 0 2 ⁢ π e 2 ⁢ i ⁢ h ⁢ w ⁢ ce n ⁡ ( t , h 2 ) ⁢ d t = i n ⁢ ce n ⁡ ( α , h 2 ) ⁢ Mc n ( 1 ) ⁡ ( z , h ) , 1 2 𝜋 superscript subscript 0 2 𝜋 superscript 𝑒 2 imaginary-unit ℎ 𝑤 Mathieu-ce 𝑛 𝑡 superscript ℎ 2 𝑡 imaginary-unit 𝑛 Mathieu-ce 𝑛 𝛼 superscript ℎ 2 modified-Mathieu-Mc 1 𝑛 𝑧 ℎ {\displaystyle{\displaystyle\dfrac{1}{2\pi}\int_{0}^{2\pi}e^{2\mathrm{i}hw}% \mathrm{ce}_{n}\left(t,h^{2}\right)\mathrm{d}t={\mathrm{i}^{n}}\mathrm{ce}_{n}% \left(\alpha,h^{2}\right){\mathrm{Mc}^{(1)}_{n}}\left(z,h\right),}}
    0 references
    DLMF id
    DLMF:28.28.E2
    0 references
    Symbols used
    Q12283
    Defining formula
    ce n ⁡ ( z , q ) Mathieu-ce 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ce}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S2.SS6.p1.m7adec
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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.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    radial Mathieu function
    Defining formula
    Mc n ( j ) ⁡ ( z , h ) modified-Mathieu-Mc 𝑗 𝑛 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{Mc}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z% },\NVar{h}\right)}}
    xml-id
    C28.S20.E15.m2adec
    0 references
    Q12294
    Defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1dec
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    Q12279
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1dec
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    Q12272
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1adec
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    Q12274
    Defining formula
    w ⁢ ( z ) 𝑤 𝑧 {\displaystyle{\displaystyle w(z)}}
    xml-id
    C28.S2.XMD1.m1adec
    0 references
     
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