DLMF:13.20.E13 (Q4602)

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DLMF:13.20.E13
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    Statements

    test
    ζ ⁢ ζ 2 - α 2 - α 2 ⁢ arccosh ⁡ ( ζ α ) = X μ - 2 ⁢ κ μ ⁢ ln ⁡ ( X + x - 2 ⁢ κ 2 ⁢ κ 2 - μ 2 ) - 2 ⁢ ln ⁡ ( κ ⁢ x - μ ⁢ X - 2 ⁢ μ 2 x ⁢ κ 2 - μ 2 ) , 𝜁 superscript 𝜁 2 superscript 𝛼 2 superscript 𝛼 2 hyperbolic-inverse-cosine 𝜁 𝛼 𝑋 𝜇 2 𝜅 𝜇 𝑋 𝑥 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 2 𝜅 𝑥 𝜇 𝑋 2 superscript 𝜇 2 𝑥 superscript 𝜅 2 superscript 𝜇 2 {\displaystyle{\displaystyle\zeta\sqrt{\zeta^{2}-\alpha^{2}}-\alpha^{2}% \operatorname{arccosh}\left(\frac{\zeta}{\alpha}\right)=\frac{X}{\mu}-\frac{2% \kappa}{\mu}\ln\left(\frac{X+x-2\kappa}{2\sqrt{\kappa^{2}-\mu^{2}}}\right)-2% \ln\left(\frac{\kappa x-\mu X-2\mu^{2}}{x\sqrt{\kappa^{2}-\mu^{2}}}\right),}}
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    DLMF id
    DLMF:13.20.E13
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    constraint
    x ≥ 2 ⁢ κ + 2 ⁢ κ 2 - μ 2 𝑥 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 {\displaystyle{\displaystyle x\geq 2\kappa+2\sqrt{\kappa^{2}-\mu^{2}}}}
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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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    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
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    Q11566
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1jdec
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    DLMF:13.20.E6
    test
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C13.S20.XMD2.m1bdec
    0 references
     
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