DLMF:19.23.E4 (Q6475)

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DLMF:19.23.E4
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    Statements

    Defining formula
    R F ⁡ ( 0 , y , z ) = 2 π ⁢ ∫ 0 π / 2 R C ⁡ ( y , z ⁢ cos 2 ⁡ θ ) ⁢ d θ = 2 π ⁢ ∫ 0 ∞ R C ⁡ ( y ⁢ cosh 2 ⁡ t , z ) ⁢ d t . Carlson-integral-RF 0 𝑦 𝑧 2 𝜋 superscript subscript 0 𝜋 2 Carlson-integral-RC 𝑦 𝑧 2 𝜃 𝜃 2 𝜋 superscript subscript 0 Carlson-integral-RC 𝑦 2 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle R_{F}\left(0,y,z\right)=\frac{2}{\pi}\int_{0}^{% \pi/2}R_{C}\left(y,z{\cos^{2}}\theta\right)\mathrm{d}\theta=\frac{2}{\pi}\int_% {0}^{\infty}R_{C}\left(y{\cosh^{2}}t,z\right)\mathrm{d}t.}}
    0 references
    DLMF id
    DLMF:19.23.E4
    0 references
    Symbols used
    Carlson’s combination of inverse circular and inverse hyperbolic functions
    Defining formula
    R C ⁡ ( x , y ) Carlson-integral-RC 𝑥 𝑦 {\displaystyle{\displaystyle R_{C}\left(\NVar{x},\NVar{y}\right)}}
    xml-id
    C19.S2.E17.m2adec
    0 references
    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.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
    hyperbolic cosine function
    Defining formula
    cosh ⁡ z 𝑧 {\displaystyle{\displaystyle\cosh\NVar{z}}}
    xml-id
    C4.S28.E2.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
     
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