DLMF:14.15.E31 (Q4879)

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DLMF:14.15.E31
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    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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