DLMF:14.15.E27 (Q4875)

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DLMF:14.15.E27
No description defined

    Statements

    test
    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),}}
    0 references
    DLMF id
    DLMF:14.15.E27
    0 references
    constraint
    α ≤ ζ < ∞ 𝛼 𝜁 {\displaystyle{\displaystyle\alpha\leq\zeta<\infty}}
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    Symbols used
    Q11246
    test
    arccosh ⁡ z hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m9adec
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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
    0 references
    Q11589
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1qdec
    0 references
    Q11620
    test
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C14.S15.XMD13.m1cdec
    0 references
    Q11621
    test
    ζ 𝜁 {\displaystyle{\displaystyle\zeta}}
    xml-id
    C14.S15.XMD14.m1bdec
    0 references
    Q11619
    test
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S15.XMD9.m1jdec
    0 references
     
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