DLMF:13.20.E15 (Q4604)

From DRMF
Jump to navigation Jump to search
No description defined
Language Label Description Also known as
English
DLMF:13.20.E15
No description defined

    Statements

    Defining formula
    - ζ ⁢ ζ 2 - α 2 - α 2 ⁢ arccosh ⁡ ( - ζ α ) = - X μ + 2 ⁢ κ μ ⁢ ln ⁡ ( 2 ⁢ κ - X - x 2 ⁢ κ 2 - μ 2 ) + 2 ⁢ ln ⁡ ( μ ⁢ X + 2 ⁢ μ 2 - κ ⁢ x 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 2 𝜅 𝑋 𝑥 2 superscript 𝜅 2 superscript 𝜇 2 2 𝜇 𝑋 2 superscript 𝜇 2 𝜅 𝑥 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{2\kappa-X-x}{2\sqrt{\kappa^{2}-\mu^{2}}}\right)+2% \ln\left(\frac{\mu X+2\mu^{2}-\kappa x}{x\sqrt{\kappa^{2}-\mu^{2}}}\right),}}
    0 references
    DLMF id
    DLMF:13.20.E15
    0 references
    constraint
    0 < x ≤ 2 ⁢ κ - 2 ⁢ κ 2 - μ 2 0 𝑥 2 𝜅 2 superscript 𝜅 2 superscript 𝜇 2 {\displaystyle{\displaystyle 0<x\leq 2\kappa-2\sqrt{\kappa^{2}-\mu^{2}}}}
    0 references
    Symbols used
    Q11246
    Defining formula
    arccosh ⁡ z hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m9aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aedec
    0 references
    Q11566
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C13.S1.XMD4.m1ldec
    0 references
    DLMF:13.20.E6
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C13.S20.XMD2.m1ddec
    0 references
    DLMF:13.20.E7
    Defining formula
    X 𝑋 {\displaystyle{\displaystyle X}}
    xml-id
    C13.S20.XMD3.m1ddec
    0 references
     
        Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Item:Q4604&oldid=63155"