DLMF:19.23.E5 (Q6476)

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DLMF:19.23.E5
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    Statements

    test
    R F ⁡ ( x , y , z ) = 2 π ⁢ ∫ 0 π / 2 R C ⁡ ( x , y ⁢ cos 2 ⁡ θ + z ⁢ sin 2 ⁡ θ ) ⁢ d θ , Carlson-integral-RF 𝑥 𝑦 𝑧 2 𝜋 superscript subscript 0 𝜋 2 Carlson-integral-RC 𝑥 𝑦 2 𝜃 𝑧 2 𝜃 𝜃 {\displaystyle{\displaystyle R_{F}\left(x,y,z\right)=\frac{2}{\pi}\int_{0}^{% \pi/2}R_{C}\left(x,y{\cos^{2}}\theta+z{\sin^{2}}\theta\right)\mathrm{d}\theta,}}
    0 references
    DLMF id
    DLMF:19.23.E5
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    constraint
    ℜ ⁡ y > 0 𝑦 0 {\displaystyle{\displaystyle\Re y>0}}
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    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    Symbols used
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    test
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2aadec
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    symmetric elliptic integral of first kind
    test
    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.m2abdec
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    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    cosine function
    test
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2addec
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    Q10770
    test
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1addec
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