DLMF:19.23.E6 (Q7173)

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DLMF:19.23.E6
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    Statements

    DLMF defining formula
    4 ⁢ π ⁢ R F ⁡ ( x , y , z ) = ∫ 0 2 ⁢ π ∫ 0 π sin ⁡ θ ⁢ d θ ⁢ d ϕ ( x ⁢ sin 2 ⁡ θ ⁢ cos 2 ⁡ ϕ + y ⁢ sin 2 ⁡ θ ⁢ sin 2 ⁡ ϕ + z ⁢ cos 2 ⁡ θ ) 1 / 2 , 4 𝜋 Carlson-integral-RF 𝑥 𝑦 𝑧 superscript subscript 0 2 𝜋 superscript subscript 0 𝜋 𝜃 𝜃 italic-ϕ superscript 𝑥 2 𝜃 2 italic-ϕ 𝑦 2 𝜃 2 italic-ϕ 𝑧 2 𝜃 1 2 {\displaystyle{\displaystyle 4\pi R_{F}\left(x,y,z\right)=\int_{0}^{2\pi}\!\!% \!\!\int_{0}^{\pi}\frac{\sin\theta\mathrm{d}\theta\mathrm{d}\phi}{(x{\sin^{2}}% \theta{\cos^{2}}\phi+y{\sin^{2}}\theta{\sin^{2}}\phi+z{\cos^{2}}\theta)^{1/2}}% ,}}
    0 references
    Symbols used
    symmetric elliptic integral of first kind
    DLMF 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.m2acdec
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    the ratio of the circumference of a circle to its diameter
    DLMF defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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    cosine function
    DLMF defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2aedec
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    differential of x
    DLMF defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
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    integral
    DLMF defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
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    sine function
    DLMF defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2addec
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    real or complex argument
    DLMF defining formula
    ϕ italic-ϕ {\displaystyle{\displaystyle\phi}}
    xml-id
    C19.S1.XMD4.m1dec
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    Identifiers

     
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