DLMF:19.2.E19 (Q6115)

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DLMF:19.2.E19
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    Statements

    test
    R C ⁡ ( x , y ) = 1 x - y ⁢ arctanh ⁡ x - y x = 1 x - y ⁢ ln ⁡ x + x - y y , Carlson-integral-RC 𝑥 𝑦 1 𝑥 𝑦 hyperbolic-inverse-tangent 𝑥 𝑦 𝑥 1 𝑥 𝑦 𝑥 𝑥 𝑦 𝑦 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=\frac{1}{\sqrt{x-y}}% \operatorname{arctanh}\sqrt{\frac{x-y}{x}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{% x}+\sqrt{x-y}}{\sqrt{y}},}}
    0 references
    DLMF id
    DLMF:19.2.E19
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    constraint
    0 < y < x 0 𝑦 𝑥 {\displaystyle{\displaystyle 0<y<x}}
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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.m2abdec
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    Q11248
    test
    arctanh ⁡ z hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m10adec
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    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
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