DLMF:10.22.E48 (Q3422): Difference between revisions

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Property / 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}}}

x>0,\realpart@@{\nu}>-1,\realpart@@{\mu}>2\realpart@@{\nu}+\tfrac{1}{2}
Property / 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}}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Bessel function of the first kind / rank
 
Normal rank
Property / Symbols used: Bessel function of the first kind / qualifier
 
test:

J ν ⁡ ( z ) Bessel-J 𝜈 𝑧 {\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}

\BesselJ{\NVar{\nu}}@{\NVar{z}}
Property / Symbols used: Bessel function of the first kind / qualifier
 
xml-id: C10.S2.E2.m2aajdec
Property / Symbols used
 
Property / Symbols used: gamma function / rank
 
Normal rank
Property / Symbols used: gamma function / qualifier
 
test:

Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}

\EulerGamma@{\NVar{z}}
Property / Symbols used: gamma function / qualifier
 
xml-id: C5.S2.E1.m2amdec
Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
test:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1aaudec
Property / Symbols used
 
Property / Symbols used: hyperbolic cosine function / rank
 
Normal rank
Property / Symbols used: hyperbolic cosine function / qualifier
 
test:

cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}

\cosh@@{\NVar{z}}
Property / Symbols used: hyperbolic cosine function / qualifier
 
xml-id: C4.S28.E2.m2adec
Property / Symbols used
 
Property / Symbols used: hyperbolic sine function / rank
 
Normal rank
Property / Symbols used: hyperbolic sine function / qualifier
 
test:

sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}

\sinh@@{\NVar{z}}
Property / Symbols used: hyperbolic sine function / qualifier
 
xml-id: C4.S28.E1.m2adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
test:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3aaudec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aacdec
Property / Symbols used
 
Property / Symbols used: Q11435 / rank
 
Normal rank
Property / Symbols used: Q11435 / qualifier
 
test:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q11435 / qualifier
 
xml-id: C10.S1.XMD4.m1qdec
Property / Symbols used
 
Property / Symbols used: Q11427 / rank
 
Normal rank
Property / Symbols used: Q11427 / qualifier
 
test:

ν 𝜈 {\displaystyle{\displaystyle\nu}}

\nu
Property / Symbols used: Q11427 / qualifier
 
xml-id: C10.S1.XMD7.m1agdec

Latest revision as of 12:21, 2 January 2020

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DLMF:10.22.E48
No description defined

    Statements

    test
    ∫ 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
    test
    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
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2amdec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aaudec
    0 references
    hyperbolic cosine function
    test
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    hyperbolic sine function
    test
    sinh ⁡ z 𝑧 {\displaystyle{\displaystyle\sinh\NVar{z}}}
    xml-id
    C4.S28.E1.m2adec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aaudec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aacdec
    0 references
    Q11435
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C10.S1.XMD4.m1qdec
    0 references
    Q11427
    test
    ν 𝜈 {\displaystyle{\displaystyle\nu}}
    xml-id
    C10.S1.XMD7.m1agdec
    0 references
     
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