DLMF:10.22.E66 (Q3440): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

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

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2afdec

Revision as of 13:26, 2 January 2020

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DLMF:10.22.E66
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    Statements

    Defining formula
    ∫ 0 ∞ e - a ⁢ t ⁢ J ν ⁡ ( b ⁢ t ) ⁢ J ν ⁡ ( c ⁢ t ) ⁢ d t = 1 π ⁢ ( b ⁢ c ) 1 2 ⁢ Q ν - 1 2 ⁡ ( a 2 + b 2 + c 2 2 ⁢ b ⁢ c ) , superscript subscript 0 superscript 𝑒 𝑎 𝑡 Bessel-J 𝜈 𝑏 𝑡 Bessel-J 𝜈 𝑐 𝑡 𝑡 1 𝜋 superscript 𝑏 𝑐 1 2 shorthand-Legendre-Q-second-kind 𝜈 1 2 superscript 𝑎 2 superscript 𝑏 2 superscript 𝑐 2 2 𝑏 𝑐 {\displaystyle{\displaystyle\int_{0}^{\infty}e^{-at}J_{\nu}\left(bt\right)J_{% \nu}\left(ct\right)\mathrm{d}t=\frac{1}{\pi(bc)^{\frac{1}{2}}}\*Q_{\nu-\frac{1% }{2}}\left(\frac{a^{2}+b^{2}+c^{2}}{2bc}\right),}}
    0 references
    DLMF id
    DLMF:10.22.E66
    0 references
    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\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.m2aaydec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aydec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1abmdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2afdec
    0 references
     
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