DLMF:28.28.E13 (Q8437)

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DLMF:28.28.E13
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    Statements

    Defining formula
    ∫ 0 ∞ e 2 ⁢ i ⁢ h ⁢ cosh ⁡ z ⁢ cosh ⁡ t ⁢ sinh ⁡ z ⁢ sinh ⁡ t ⁢ Fe m ⁡ ( t , h 2 ) ⁢ d t = - π 4 ⁢ h ⁢ i m ⁢ fe m ′ ⁡ ( 0 , h 2 ) ⁢ Mc m ( 3 ) ⁡ ( z , h ) , superscript subscript 0 superscript 𝑒 2 imaginary-unit ℎ 𝑧 𝑡 𝑧 𝑡 modified-Mathieu-Fe 𝑚 𝑡 superscript ℎ 2 𝑡 𝜋 4 ℎ imaginary-unit 𝑚 diffop Mathieu-fe 𝑚 1 0 superscript ℎ 2 modified-Mathieu-Mc 3 𝑚 𝑧 ℎ {\displaystyle{\displaystyle\int_{0}^{\infty}e^{2\mathrm{i}h\cosh z\cosh t}% \sinh z\sinh t\mathrm{Fe}_{m}\left(t,h^{2}\right)\mathrm{d}t=-\dfrac{\pi}{4h}{% \mathrm{i}^{m}}\mathrm{fe}_{m}'\left(0,h^{2}\right){\mathrm{Mc}^{(3)}_{m}}% \left(z,h\right),}}
    0 references
    DLMF id
    DLMF:28.28.E13
    0 references
    Symbols used
    second solution, Mathieu’s equation
    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.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.m2akdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aidec
    0 references
     
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