DLMF:14.15.E27 (Q4875)

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DLMF:14.15.E27
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    1 2 ζ ( ζ 2 - α 2 ) 1 / 2 - 1 2 α 2 arccosh ( ζ α ) = ( 1 - a 2 ) 1 / 2 arctanh ( 1 x ( x 2 - a 2 1 - a 2 ) 1 / 2 ) - arccosh ( x a ) , 1 2 𝜁 superscript superscript 𝜁 2 superscript 𝛼 2 1 2 1 2 superscript 𝛼 2 hyperbolic-inverse-cosine 𝜁 𝛼 superscript 1 superscript 𝑎 2 1 2 hyperbolic-inverse-tangent 1 𝑥 superscript superscript 𝑥 2 superscript 𝑎 2 1 superscript 𝑎 2 1 2 hyperbolic-inverse-cosine 𝑥 𝑎 {\displaystyle{\displaystyle\frac{1}{2}\zeta\left(\zeta^{2}-\alpha^{2}\right)^% {1/2}-\frac{1}{2}\alpha^{2}\operatorname{arccosh}\left(\frac{\zeta}{\alpha}% \right)=\left(1-a^{2}\right)^{1/2}\operatorname{arctanh}\left(\frac{1}{x}\left% (\frac{x^{2}-a^{2}}{1-a^{2}}\right)^{1/2}\right)-\operatorname{arccosh}\left(% \frac{x}{a}\right),}}
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    DLMF:14.15.E27
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