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DLMF:28.28.E32 - MaRDI portal

DLMF:28.28.E32 (Q9151)

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DLMF:28.28.E32
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    Statements

    DLMF defining formula
    sinh ⁡ ( 2 ⁢ z ) π 2 ⁢ ∫ 0 2 ⁢ π me ν ′ ⁡ ( t , h 2 ) ⁢ me - ν - 2 ⁢ m ⁡ ( t , h 2 ) sinh 2 ⁡ z + sin 2 ⁡ t ⁢ d t = ( - 1 ) m + 1 ⁢ i ⁢ γ ν , m ⁢ D 1 ⁡ ( ν , ν + 2 ⁢ m , z ) , 2 𝑧 superscript 𝜋 2 superscript subscript 0 2 𝜋 diffop Mathieu-me 𝜈 1 𝑡 superscript ℎ 2 Mathieu-me 𝜈 2 𝑚 𝑡 superscript ℎ 2 2 𝑧 2 𝑡 𝑡 superscript 1 𝑚 1 imaginary-unit subscript 𝛾 𝜈 𝑚 Mathieu-D 1 𝜈 𝜈 2 𝑚 𝑧 {\displaystyle{\displaystyle\dfrac{\sinh\left(2z\right)}{\pi^{2}}\int_{0}^{2% \pi}\dfrac{\mathrm{me}_{\nu}'\left(t,h^{2}\right)\mathrm{me}_{-\nu-2m}\left(t,% h^{2}\right)}{{\sinh^{2}}z+{\sin^{2}}t}\mathrm{d}t=(-1)^{m+1}\mathrm{i}\gamma_% {\nu,m}\mathrm{D}_{1}\left(\nu,\nu+2m,z\right),}}
    0 references
    Symbols used
    cross-products of modified Mathieu functions and their derivatives
    DLMF defining formula
    D j ⁡ ( ν , μ , z ) Mathieu-D 𝑗 𝜈 𝜇 𝑧 {\displaystyle{\displaystyle\mathrm{D}_{\NVar{j}}\left(\NVar{\nu},\NVar{\mu},% \NVar{z}\right)}}
    xml-id
    C28.S28.E24.m1ahdec
    0 references
    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.m2akdec
    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.m2aaadec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aydec
    0 references
    hyperbolic sine function
    DLMF defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2ahdec
    0 references
    imaginary unit
    DLMF defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2audec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aydec
    0 references
    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2ajdec
    0 references
    integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C28.S1.XMD1.m1mdec
    0 references
    parameter
    DLMF defining formula
    h ℎ {\displaystyle{\displaystyle h}}
    xml-id
    C28.S1.XMD11.m1aadec
    0 references
    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C28.S1.XMD6.m1zdec
    0 references
    complex parameter
    DLMF defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C28.S1.XMD7.m1ndec
    0 references
    28.28.E33
    DLMF defining formula
    γ ν , m subscript 𝛾 𝜈 𝑚 {\displaystyle{\displaystyle\gamma_{\nu,m}}}
    xml-id
    C28.S28.XMD8.m1adec
    0 references

    Identifiers

     
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