DLMF:19.8.E4 (Q6210)

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DLMF:19.8.E4
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    Statements

    Defining formula
    1 M ⁡ ( a 0 , g 0 ) = 2 π ⁢ ∫ 0 π / 2 d θ a 0 2 ⁢ cos 2 ⁡ θ + g 0 2 ⁢ sin 2 ⁡ θ = 1 π ⁢ ∫ 0 ∞ d t t ⁢ ( t + a 0 2 ) ⁢ ( t + g 0 2 ) . 1 arithmetic-geometric-mean subscript 𝑎 0 subscript 𝑔 0 2 𝜋 superscript subscript 0 𝜋 2 𝜃 superscript subscript 𝑎 0 2 2 𝜃 superscript subscript 𝑔 0 2 2 𝜃 1 𝜋 superscript subscript 0 𝑡 𝑡 𝑡 superscript subscript 𝑎 0 2 𝑡 superscript subscript 𝑔 0 2 {\displaystyle{\displaystyle\frac{1}{M\left(a_{0},g_{0}\right)}=\frac{2}{\pi}% \int_{0}^{\pi/2}\frac{\mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos^{2}}\theta+g_{0}^% {2}{\sin^{2}}\theta}}=\frac{1}{\pi}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t% (t+a_{0}^{2})(t+g_{0}^{2})}}.}}
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    DLMF id
    DLMF:19.8.E4
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    Symbols used
    Q11826
    Defining formula
    M ⁡ ( a , g ) arithmetic-geometric-mean 𝑎 𝑔 {\displaystyle{\displaystyle M\left(\NVar{a},\NVar{g}\right)}}
    xml-id
    C19.S8.SS1.p1.m7adec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
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    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2adec
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