DLMF:19.28.E10 (Q6618)

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DLMF:19.28.E10
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    Statements

    Defining formula
    ∫ 0 ∞ R F ⁡ ( ( a ⁢ c + b ⁢ d ) 2 , ( a ⁢ d + b ⁢ c ) 2 , 4 ⁢ a ⁢ b ⁢ c ⁢ d ⁢ cosh 2 ⁡ z ) ⁢ d z = 1 2 ⁢ R F ⁡ ( 0 , a 2 , b 2 ) ⁢ R F ⁡ ( 0 , c 2 , d 2 ) , superscript subscript 0 Carlson-integral-RF superscript 𝑎 𝑐 𝑏 𝑑 2 superscript 𝑎 𝑑 𝑏 𝑐 2 4 𝑎 𝑏 𝑐 𝑑 2 𝑧 𝑧 1 2 Carlson-integral-RF 0 superscript 𝑎 2 superscript 𝑏 2 Carlson-integral-RF 0 superscript 𝑐 2 superscript 𝑑 2 {\displaystyle{\displaystyle\int_{0}^{\infty}R_{F}\left((ac+bd)^{2},(ad+bc)^{2% },4abcd{\cosh^{2}}z\right)\mathrm{d}z=\tfrac{1}{2}R_{F}\left(0,a^{2},b^{2}% \right)R_{F}\left(0,c^{2},d^{2}\right),}}
    0 references
    DLMF id
    DLMF:19.28.E10
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    constraint
    a , b , c , d > 0 𝑎 𝑏 𝑐 𝑑 0 {\displaystyle{\displaystyle a,b,c,d>0}}
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    Symbols used
    symmetric elliptic integral of first kind
    Defining formula
    R F ⁡ ( x , y , z ) Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)}}
    xml-id
    C19.S16.E1.m2aedec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aidec
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