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DLMF:28.28.E7 - MaRDI portal

DLMF:28.28.E7 (Q9123)

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DLMF:28.28.E7
No description defined

    Statements

    DLMF defining formula
    1 π ⁢ ∫ ℒ j e 2 ⁢ i ⁢ h ⁢ w ⁢ me ν ⁡ ( t , h 2 ) ⁢ d t = e i ⁢ ν ⁢ π / 2 ⁢ me ν ⁡ ( α , h 2 ) ⁢ M ν ( j ) ⁡ ( z , h ) , 1 𝜋 subscript subscript ℒ 𝑗 superscript 𝑒 2 imaginary-unit ℎ 𝑤 Mathieu-me 𝜈 𝑡 superscript ℎ 2 𝑡 superscript 𝑒 imaginary-unit 𝜈 𝜋 2 Mathieu-me 𝜈 𝛼 superscript ℎ 2 modified-Mathieu-M 𝑗 𝜈 𝑧 ℎ {\displaystyle{\displaystyle\dfrac{1}{\pi}\int_{\mathcal{L}_{j}}e^{2\mathrm{i}% hw}\mathrm{me}_{\nu}\left(t,h^{2}\right)\mathrm{d}t=e^{\mathrm{i}\nu\pi/{2}}% \mathrm{me}_{\nu}\left(\alpha,h^{2}\right){\mathrm{M}^{(j)}_{\nu}}\left(z,h% \right),}}
    0 references
    constraint
    j = 3 , 4 𝑗 3 4 {\displaystyle{\displaystyle j=3,4}}
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    Symbols used
    Q12302
    DLMF 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.m2adec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
    0 references
    base of natural logarithm
    DLMF defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2addec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3addec
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    Q12304
    DLMF 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.m4adec
    0 references
    parameter
    DLMF defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1ddec
    0 references
    integer
    DLMF defining formula
    j 𝑗 {\displaystyle{\displaystyle j}}
    xml-id
    C28.S1.XMD3.m1adec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1edec
    0 references
    complex parameter
    DLMF defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C28.S1.XMD7.m1dec
    0 references
    Mathieu’s equation solution
    DLMF defining formula
    w ⁢ ( z ) 𝑤 𝑧 {\displaystyle{\displaystyle w(z)}}
    xml-id
    C28.S2.XMD1.m1edec
    0 references
    parameter
    DLMF defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C28.S28.XMD1.m1edec
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    integration paths
    DLMF defining formula
    ℒ j subscript ℒ 𝑗 {\displaystyle{\displaystyle\mathcal{L}_{j}}}
    xml-id
    C28.S28.XMD2.m1adec
    0 references

    Identifiers

     
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