DLMF:10.22.E48 (Q3422)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:10.22.E48
No description defined

    Statements

    Defining formula
    ∫ 0 ∞ J μ ⁡ ( x ⁢ cosh ⁡ ϕ ) ⁢ ( cosh ⁡ ϕ ) 1 - μ ⁢ ( sinh ⁡ ϕ ) 2 ⁢ ν + 1 ⁢ d ϕ = 2 ν ⁢ Γ ⁡ ( ν + 1 ) ⁢ x - ν - 1 ⁢ J μ - ν - 1 ⁡ ( x ) , superscript subscript 0 Bessel-J 𝜇 𝑥 italic-ϕ superscript italic-ϕ 1 𝜇 superscript italic-ϕ 2 𝜈 1 italic-ϕ superscript 2 𝜈 Euler-Gamma 𝜈 1 superscript 𝑥 𝜈 1 Bessel-J 𝜇 𝜈 1 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}J_{\mu}\left(x\cosh\phi\right)(% \cosh\phi)^{1-\mu}(\sinh\phi)^{2\nu+1}\mathrm{d}\phi=2^{\nu}\Gamma\left(\nu+1% \right)x^{-\nu-1}J_{\mu-\nu-1}\left(x\right),}}
    0 references
    DLMF id
    DLMF:10.22.E48
    0 references
    constraint
    x > 0 , ℜ ⁡ ν > - 1 , ℜ ⁡ μ > 2 ⁢ ℜ ⁡ ν + 1 2 formulae-sequence 𝑥 0 formulae-sequence 𝜈 1 𝜇 2 𝜈 1 2 {\displaystyle{\displaystyle x>0,\Re\nu>-1,\Re\mu>2\Re\nu+\tfrac{1}{2}}}
    0 references
    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.m2aajdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2amdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaudec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    hyperbolic sine function
    Defining formula
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaudec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aacdec
    0 references
    Q11435
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1qdec
    0 references
    Q11427
    Defining formula
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1agdec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q3422&oldid=35801"