DLMF:19.2.E20 (Q6116): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
 
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2aadec

Latest revision as of 13:36, 2 January 2020

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DLMF:19.2.E20
No description defined

    Statements

    Defining formula
    R C ⁡ ( x , y ) = x x - y ⁢ R C ⁡ ( x - y , - y ) = 1 x - y ⁢ arctanh ⁡ x x - y = 1 x - y ⁢ ln ⁡ x + x - y - y , Carlson-integral-RC 𝑥 𝑦 𝑥 𝑥 𝑦 Carlson-integral-RC 𝑥 𝑦 𝑦 1 𝑥 𝑦 hyperbolic-inverse-tangent 𝑥 𝑥 𝑦 1 𝑥 𝑦 𝑥 𝑥 𝑦 𝑦 {\displaystyle{\displaystyle R_{C}\left(x,y\right)=\sqrt{\frac{x}{x-y}}R_{C}% \left(x-y,-y\right)=\frac{1}{\sqrt{x-y}}\operatorname{arctanh}\sqrt{\frac{x}{x% -y}}=\frac{1}{\sqrt{x-y}}\ln\frac{\sqrt{x}+\sqrt{x-y}}{\sqrt{-y}},}}
    0 references
    DLMF id
    DLMF:19.2.E20
    0 references
    constraint
    y < 0 ≤ x 𝑦 0 𝑥 {\displaystyle{\displaystyle y<0\leq x}}
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    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.m2acdec
    0 references
    Q11248
    Defining formula
    arctanh ⁡ z hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m10aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
     
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