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DLMF:15.9.E12 - MaRDI portal

DLMF:15.9.E12 (Q5799)

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DLMF:15.9.E12
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    Statements

    DLMF defining formula
    f ~ ⁢ ( λ ) = ∫ 0 ∞ f ⁢ ( t ) ⁢ ϕ λ ( α , β ) ⁡ ( t ) ⁢ ( 2 ⁢ sinh ⁡ t ) 2 ⁢ α + 1 ⁢ ( 2 ⁢ cosh ⁡ t ) 2 ⁢ β + 1 ⁢ d t , ~ 𝑓 𝜆 superscript subscript 0 𝑓 𝑡 Jacobi-hypergeometric-phi 𝛼 𝛽 𝜆 𝑡 superscript 2 𝑡 2 𝛼 1 superscript 2 𝑡 2 𝛽 1 𝑡 {\displaystyle{\displaystyle\widetilde{f}(\lambda)=\int_{0}^{\infty}f(t)\phi^{% (\alpha,\beta)}_{\lambda}\left(t\right)(2\sinh t)^{2\alpha+1}(2\cosh t)^{2% \beta+1}\mathrm{d}t,}}
    0 references
    Symbols used
    Jacobi function
    DLMF defining formula
    ϕ λ ( α , β ) ⁡ ( t ) Jacobi-hypergeometric-phi 𝛼 𝛽 𝜆 𝑡 {\displaystyle{\displaystyle\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda% }}\left(\NVar{t}\right)}}
    xml-id
    C15.S9.E11.m2aadec
    0 references
    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    hyperbolic cosine function
    DLMF defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    hyperbolic sine function
    DLMF defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2aadec
    0 references
    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    function
    DLMF defining formula
    f ⁢ ( t ) 𝑓 𝑡 {\displaystyle{\displaystyle f(t)}}
    xml-id
    C15.S9.XMD1.m1dec
    0 references

    Identifiers

     
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