DLMF:13.23.E11 (Q4645): Difference between revisions

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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.m2ajdec
Property / Symbols used
 
Property / Symbols used: Whittaker confluent hypergeometric function / rank
 
Normal rank
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
test:

W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}

\WhittakerconfhyperW{\NVar{\kappa}}{\NVar{\mu}}@{\NVar{z}}
Property / Symbols used: Whittaker confluent hypergeometric function / qualifier
 
xml-id: C13.S14.E3.m2aedec
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.m1ajdec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
test:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2ajdec
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.m3ajdec
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.m1ajdec

Latest revision as of 00:04, 2 January 2020

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English
DLMF:13.23.E11
No description defined

    Statements

    test
    ∫ 0 ∞ e 1 2 ⁢ t ⁢ t 1 2 ⁢ ( ν - 1 ) - μ ⁢ W κ , μ ⁡ ( t ) ⁢ J ν ⁡ ( 2 ⁢ x ⁢ t ) ⁢ d t = Γ ⁡ ( ν - 2 ⁢ μ + 1 ) Γ ⁡ ( 1 2 + μ - κ ) ⁢ e 1 2 ⁢ x ⁢ x 1 2 ⁢ ( μ - κ - 3 2 ) ⁢ W 1 2 ⁢ ( κ + 3 ⁢ μ - ν - 1 2 ) , 1 2 ⁢ ( κ - μ + ν + 1 2 ) ⁡ ( x ) , superscript subscript 0 superscript 𝑒 1 2 𝑡 superscript 𝑡 1 2 𝜈 1 𝜇 Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑡 Bessel-J 𝜈 2 𝑥 𝑡 𝑡 Euler-Gamma 𝜈 2 𝜇 1 Euler-Gamma 1 2 𝜇 𝜅 superscript 𝑒 1 2 𝑥 superscript 𝑥 1 2 𝜇 𝜅 3 2 Whittaker-confluent-hypergeometric-W 1 2 𝜅 3 𝜇 𝜈 1 2 1 2 𝜅 𝜇 𝜈 1 2 𝑥 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{\frac{1}{2}t}t^{\frac{1}{2}(% \nu-1)-\mu}W_{\kappa,\mu}\left(t\right)J_{\nu}\left(2\sqrt{xt}\right)\mathrm{d% }t=\frac{\Gamma\left(\nu-2\mu+1\right)}{\Gamma\left(\frac{1}{2}+\mu-\kappa% \right)}\*e^{\frac{1}{2}x}x^{\frac{1}{2}(\mu-\kappa-\frac{3}{2})}\*W_{\frac{1}% {2}(\kappa+3\mu-\nu-\frac{1}{2}),\frac{1}{2}(\kappa-\mu+\nu+\frac{1}{2})}\left% (x\right),}}
    0 references
    DLMF id
    DLMF:13.23.E11
    0 references
    constraint
    max ⁡ ( 2 ⁢ ℜ ⁡ μ - 1 , - 1 ) < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ ( μ - κ ) + 3 2 2 𝜇 1 1 𝜈 2 𝜇 𝜅 3 2 {\displaystyle{\displaystyle\max(2\Re\mu-1,-1)<\Re\nu<2\Re(\mu-\kappa)+\tfrac{% 3}{2}}}
    0 references
    x > 0 𝑥 0 {\displaystyle{\displaystyle x>0}}
    0 references
    max ⁡ ( 2 ⁢ ℜ ⁡ μ - 1 , - 1 ) < ℜ ⁡ ν < 2 ⁢ ℜ ⁡ ( μ - κ ) + 3 2 2 𝜇 1 1 𝜈 2 𝜇 𝜅 3 2 {\displaystyle{\displaystyle\max(2\Re\mu-1,-1)<\Re\nu<2\Re(\mu-\kappa)+\tfrac{% 3}{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.m2abdec
    0 references
    gamma function
    test
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2ajdec
    0 references
    Whittaker confluent hypergeometric function
    test
    W κ , μ ⁡ ( z ) Whittaker-confluent-hypergeometric-W 𝜅 𝜇 𝑧 {\displaystyle{\displaystyle W_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)}}
    xml-id
    C13.S14.E3.m2aedec
    0 references
    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1ajdec
    0 references
    base of natural logarithm
    test
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2ajdec
    0 references
    Q10771
    test
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3ajdec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1ajdec
    0 references
     
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