DLMF:10.22.E25 (Q3399)

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DLMF:10.22.E25
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    Statements

    Defining formula
    ∫ 0 1 2 ⁢ π J μ ⁡ ( z ⁢ sin ⁡ θ ) ⁢ I ν ⁡ ( z ⁢ cos ⁡ θ ) ⁢ ( tan ⁡ θ ) μ + 1 ⁢ d θ = Γ ⁡ ( 1 2 ⁢ ν - 1 2 ⁢ μ ) ⁢ ( 1 2 ⁢ z ) μ 2 ⁢ Γ ⁡ ( 1 2 ⁢ ν + 1 2 ⁢ μ + 1 ) ⁢ J ν ⁡ ( z ) , superscript subscript 0 1 2 𝜋 Bessel-J 𝜇 𝑧 𝜃 modified-Bessel-first-kind 𝜈 𝑧 𝜃 superscript 𝜃 𝜇 1 𝜃 Euler-Gamma 1 2 𝜈 1 2 𝜇 superscript 1 2 𝑧 𝜇 2 Euler-Gamma 1 2 𝜈 1 2 𝜇 1 Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle\int_{0}^{\frac{1}{2}\pi}J_{\mu}\left(z\sin\theta% \right)I_{\nu}\left(z\cos\theta\right)(\tan\theta)^{\mu+1}\mathrm{d}\theta=% \frac{\Gamma\left(\tfrac{1}{2}\nu-\tfrac{1}{2}\mu\right)(\tfrac{1}{2}z)^{\mu}}% {2\Gamma\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1\right)}J_{\nu}\left(z\right),}}
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    DLMF id
    DLMF:10.22.E25
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    constraint
    ℜ ⁡ ν > ℜ ⁡ μ > - 1 𝜈 𝜇 1 {\displaystyle{\displaystyle\Re\nu>\Re\mu>-1}}
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    Symbols used
    Bessel function of the first kind
    Defining formula
    J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E2.m2aqdec
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    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2agdec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2amdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2akdec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1axdec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3axdec
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    modified Bessel function of the first kind
    Defining formula
    I ν ⁡ ( z ) modified-Bessel-first-kind 𝜈 𝑧 {\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}
    xml-id
    C10.S25.E2.m2adec
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    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aldec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2ajdec
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    tangent function
    Defining formula
    tan ⁡ z 𝑧 {\displaystyle{\displaystyle\tan\NVar{z}}}
    xml-id
    C4.S14.E4.m2adec
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    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1sdec
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    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1odec
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