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DLMF:28.28.E16 - MaRDI portal

DLMF:28.28.E16 (Q9132)

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DLMF:28.28.E16
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    Statements

    DLMF defining formula
    ∫ 0 ∞ sin ⁡ ( 2 ⁢ h ⁢ cos ⁡ y ⁢ cosh ⁡ t ) ⁢ Ce 2 ⁢ n ⁡ ( t , h 2 ) ⁢ d t = - π ⁢ A 0 2 ⁢ n ⁢ ( h 2 ) 2 ⁢ ce 2 ⁢ n ⁡ ( 1 2 ⁢ π , h 2 ) ⁢ ( ce 2 ⁢ n ⁡ ( y , h 2 ) ∓ 2 π ⁢ C 2 ⁢ n ⁢ ( h 2 ) ⁢ fe 2 ⁢ n ⁡ ( y , h 2 ) ) , superscript subscript 0 2 ℎ 𝑦 𝑡 modified-Mathieu-Ce 2 𝑛 𝑡 superscript ℎ 2 𝑡 𝜋 superscript subscript 𝐴 0 2 𝑛 superscript ℎ 2 2 Mathieu-ce 2 𝑛 1 2 𝜋 superscript ℎ 2 minus-or-plus Mathieu-ce 2 𝑛 𝑦 superscript ℎ 2 2 𝜋 subscript 𝐶 2 𝑛 superscript ℎ 2 Mathieu-fe 2 𝑛 𝑦 superscript ℎ 2 {\displaystyle{\displaystyle\int_{0}^{\infty}\sin\left(2h\cos y\cosh t\right)% \mathrm{Ce}_{2n}\left(t,h^{2}\right)\mathrm{d}t=-\dfrac{\pi A_{0}^{2n}(h^{2})}% {2\mathrm{ce}_{2n}\left(\frac{1}{2}\pi,h^{2}\right)}\*\left(\mathrm{ce}_{2n}% \left(y,h^{2}\right)\mp\dfrac{2}{\pi C_{2n}(h^{2})}\mathrm{fe}_{2n}\left(y,h^{% 2}\right)\right),}}
    0 references
    Symbols used
    Q12286
    DLMF 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.m7addec
    0 references
    second solution, Mathieu’s equation
    DLMF defining formula
    fe n ⁡ ( z , q ) Mathieu-fe 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{fe}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S5.E1.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2andec
    0 references
    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2abdec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aldec
    0 references
    hyperbolic cosine function
    DLMF defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2agdec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aldec
    0 references
    modified Mathieu function
    DLMF defining formula
    Ce ν ⁡ ( z , q ) modified-Mathieu-Ce 𝜈 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{Ce}_{\NVar{\nu}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S20.E3.m2abdec
    0 references
    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2aadec
    0 references
    parameter
    DLMF defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1mdec
    0 references
    integer
    DLMF defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C28.S1.XMD2.m1edec
    0 references
    real variable
    DLMF defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C28.S1.XMD5.m1adec
    0 references
    Fourier coefficient
    DLMF defining formula
    A m ⁢ ( q ) subscript 𝐴 𝑚 𝑞 {\displaystyle{\displaystyle A_{m}(q)}}
    xml-id
    C28.S4.XMD1.m1dec
    0 references
    factor
    DLMF defining formula
    C n ⁢ ( q ) subscript 𝐶 𝑛 𝑞 {\displaystyle{\displaystyle C_{n}(q)}}
    xml-id
    C28.S5.XMD3.m1dec
    0 references

    Identifiers

     
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