DLMF:14.15.E31 (Q4879)

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DLMF:14.15.E31
No description defined

    Statements

    Defining formula
    1 2 ⁢ ζ ⁢ ( ζ 2 + α 2 ) 1 / 2 + 1 2 ⁢ α 2 ⁢ arcsinh ⁡ ( ζ α ) = ( 1 + a 2 ) 1 / 2 ⁢ arctanh ⁡ ( x ⁢ ( 1 + a 2 x 2 + a 2 ) 1 / 2 ) - arcsinh ⁡ ( x a ) , 1 2 𝜁 superscript superscript 𝜁 2 superscript 𝛼 2 1 2 1 2 superscript 𝛼 2 hyperbolic-inverse-sine 𝜁 𝛼 superscript 1 superscript 𝑎 2 1 2 hyperbolic-inverse-tangent 𝑥 superscript 1 superscript 𝑎 2 superscript 𝑥 2 superscript 𝑎 2 1 2 hyperbolic-inverse-sine 𝑥 𝑎 {\displaystyle{\displaystyle\frac{1}{2}\zeta\left(\zeta^{2}+\alpha^{2}\right)^% {1/2}+\frac{1}{2}\alpha^{2}\operatorname{arcsinh}\left(\frac{\zeta}{\alpha}% \right)=\left(1+a^{2}\right)^{1/2}\operatorname{arctanh}\left(x\left(\frac{1+a% ^{2}}{x^{2}+a^{2}}\right)^{1/2}\right)-\operatorname{arcsinh}\left(\frac{x}{a}% \right),}}
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    DLMF id
    DLMF:14.15.E31
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    constraint
    - ∞ < ζ < ∞ 𝜁 {\displaystyle{\displaystyle-\infty<\zeta<\infty}}
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    Symbols used
    Q11247
    Defining formula
    arcsinh ⁡ z hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m8adec
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    Q11248
    Defining formula
    arctanh ⁡ z hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m10aadec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1udec
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    Q11620
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C14.S15.XMD13.m1fdec
    0 references
    Q11621
    Defining formula
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C14.S15.XMD14.m1fdec
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    Q11619
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD9.m1mdec
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