DLMF:28.28.E14 (Q8438)

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DLMF:28.28.E14
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    Statements

    Defining formula
    ∫ 0 ∞ e 2 ⁢ i ⁢ h ⁢ cosh ⁡ z ⁢ cosh ⁡ t ⁢ Ge m ⁡ ( t , h 2 ) ⁢ d t = 1 2 ⁢ π ⁢ i m + 1 ⁢ ge m ⁡ ( 0 , h 2 ) ⁢ Ms m ( 3 ) ⁡ ( z , h ) . superscript subscript 0 superscript 𝑒 2 imaginary-unit ℎ 𝑧 𝑡 modified-Mathieu-Ge 𝑚 𝑡 superscript ℎ 2 𝑡 1 2 𝜋 imaginary-unit 𝑚 1 Mathieu-ge 𝑚 0 superscript ℎ 2 modified-Mathieu-Ms 3 𝑚 𝑧 ℎ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{2\mathrm{i}h\cosh z\cosh t}% \mathrm{Ge}_{m}\left(t,h^{2}\right)\mathrm{d}t=\tfrac{1}{2}\pi{\mathrm{i}^{m+1% }}\mathrm{ge}_{m}\left(0,h^{2}\right){\mathrm{Ms}^{(3)}_{m}}\left(z,h\right).}}
    0 references
    DLMF id
    DLMF:28.28.E14
    0 references
    Symbols used
    second solution, Mathieu’s equation
    Defining formula
    ge n ⁡ ( z , q ) Mathieu-ge 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S5.E2.m2adec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aldec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2aedec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2akdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
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    modified Mathieu function
    Defining formula
    Ge n ⁡ ( z , q ) modified-Mathieu-Ge 𝑛 𝑧 𝑞 {\displaystyle{\displaystyle\mathrm{Ge}_{\NVar{n}}\left(\NVar{z},\NVar{q}% \right)}}
    xml-id
    C28.S20.E7.m2adec
    0 references
    radial Mathieu function
    Defining formula
    Ms n ( j ) ⁡ ( z , h ) modified-Mathieu-Ms 𝑗 𝑛 𝑧 ℎ {\displaystyle{\displaystyle{\mathrm{Ms}^{(\NVar{j})}_{\NVar{n}}}\left(\NVar{z% },\NVar{h}\right)}}
    xml-id
    C28.S20.E16.m2abdec
    0 references
    Q12284
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1adec
    0 references
     
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