DLMF:14.20.E24 (Q4945)

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DLMF:14.20.E24
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    Statements

    Defining formula
    ρ = 1 2 ⁢ ln ⁡ ( ( 1 - β 2 ) ⁢ x 2 + 1 + β 2 + 2 ⁢ x ⁢ ( 1 + β 2 - β 2 ⁢ x 2 ) 1 / 2 1 - x 2 ) + β ⁢ arctan ⁡ ( β ⁢ x 1 + β 2 - β 2 ⁢ x 2 ) - 1 2 ⁢ ln ⁡ ( 1 + β 2 ) , 𝜌 1 2 1 superscript 𝛽 2 superscript 𝑥 2 1 superscript 𝛽 2 2 𝑥 superscript 1 superscript 𝛽 2 superscript 𝛽 2 superscript 𝑥 2 1 2 1 superscript 𝑥 2 𝛽 𝛽 𝑥 1 superscript 𝛽 2 superscript 𝛽 2 superscript 𝑥 2 1 2 1 superscript 𝛽 2 {\displaystyle{\displaystyle\rho=\frac{1}{2}\ln\left(\frac{\left(1-\beta^{2}% \right)x^{2}+1+\beta^{2}+2x\left(1+\beta^{2}-\beta^{2}x^{2}\right)^{1/2}}{1-x^% {2}}\right)+\beta\operatorname{arctan}\left(\frac{\beta x}{\sqrt{1+\beta^{2}-% \beta^{2}x^{2}}}\right)-\frac{1}{2}\ln\left(1+\beta^{2}\right),}}
    0 references
    DLMF id
    DLMF:14.20.E24
    0 references
    Symbols used
    Q10816
    Defining formula
    arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7abdec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
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