DLMF:14.15.E22 (Q4868)

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DLMF:14.15.E22
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    Statements

    Defining formula
    ( α 2 - y ) 1 / 2 + 1 2 ⁢ α ⁢ ln ⁡ | y | - α ⁢ ln ⁡ ( ( α 2 - y ) 1 / 2 + α ) = ln ⁡ ( x + ( x 2 - 1 + α 2 ) 1 / 2 ( 1 - α 2 ) 1 / 2 ) + α 2 ⁢ ln ⁡ ( ( 1 - α 2 ) ⁢ | 1 - x 2 | ( 1 + α 2 ) ⁢ x 2 - 1 + α 2 + 2 ⁢ α ⁢ x ⁢ ( x 2 - 1 + α 2 ) 1 / 2 ) , superscript superscript 𝛼 2 𝑦 1 2 1 2 𝛼 𝑦 𝛼 superscript superscript 𝛼 2 𝑦 1 2 𝛼 𝑥 superscript superscript 𝑥 2 1 superscript 𝛼 2 1 2 superscript 1 superscript 𝛼 2 1 2 𝛼 2 1 superscript 𝛼 2 1 superscript 𝑥 2 1 superscript 𝛼 2 superscript 𝑥 2 1 superscript 𝛼 2 2 𝛼 𝑥 superscript superscript 𝑥 2 1 superscript 𝛼 2 1 2 {\displaystyle{\displaystyle{\left(\alpha^{2}-y\right)^{1/2}+\tfrac{1}{2}% \alpha\ln|y|-\alpha\ln\left(\left(\alpha^{2}-y\right)^{1/2}+\alpha\right)}={% \ln\left(\frac{x+\left(x^{2}-1+\alpha^{2}\right)^{1/2}}{\left(1-\alpha^{2}% \right)^{1/2}}\right)+\frac{\alpha}{2}\ln\left(\frac{\left(1-\alpha^{2}\right)% \left|1-x^{2}\right|}{\left(1+\alpha^{2}\right)x^{2}-1+\alpha^{2}+2\alpha x% \left(x^{2}-1+\alpha^{2}\right)^{1/2}}\right)},}}
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    DLMF id
    DLMF:14.15.E22
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    constraint
    y ≤ α 2 𝑦 superscript 𝛼 2 {\displaystyle{\displaystyle y\leq\alpha^{2}}}
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    Symbols used
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
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    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1mdec
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