DLMF:14.20.E21 (Q4942)

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DLMF:14.20.E21
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    Statements

    Defining formula
    ( α 2 + η ) 1 / 2 + 1 2 ⁢ α ⁢ ln ⁡ η - α ⁢ ln ⁡ ( ( α 2 + η ) 1 / 2 + α ) = arccos ⁡ ( x ( 1 + α 2 ) 1 / 2 ) + α 2 ⁢ ln ⁡ ( 1 + α 2 + ( α 2 - 1 ) ⁢ x 2 - 2 ⁢ α ⁢ x ⁢ ( 1 + α 2 - x 2 ) 1 / 2 ( 1 + α 2 ) ⁢ ( 1 - x 2 ) ) , superscript superscript 𝛼 2 𝜂 1 2 1 2 𝛼 𝜂 𝛼 superscript superscript 𝛼 2 𝜂 1 2 𝛼 𝑥 superscript 1 superscript 𝛼 2 1 2 𝛼 2 1 superscript 𝛼 2 superscript 𝛼 2 1 superscript 𝑥 2 2 𝛼 𝑥 superscript 1 superscript 𝛼 2 superscript 𝑥 2 1 2 1 superscript 𝛼 2 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\left(\alpha^{2}+\eta\right)^{1/2}+\tfrac{1}{2}% \alpha\ln\eta-\alpha\ln\left(\left(\alpha^{2}+\eta\right)^{1/2}+\alpha\right)}% ={\operatorname{arccos}\left(\frac{x}{\left(1+\alpha^{2}\right)^{1/2}}\right)+% \frac{\alpha}{2}\ln\left(\frac{1+\alpha^{2}+\left(\alpha^{2}-1\right)x^{2}-2% \alpha x\left(1+\alpha^{2}-x^{2}\right)^{1/2}}{\left(1+\alpha^{2}\right)\left(% 1-x^{2}\right)}\right)},}}
    0 references
    DLMF id
    DLMF:14.20.E21
    0 references
    Symbols used
    Q11208
    Defining formula
    arccos ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m6adec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1vdec
    0 references
    Q11633
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C14.S20.XMD5.m1ddec
    0 references
    Q11635
    Defining formula
    η 𝜂 {\displaystyle{\displaystyle\eta}}
    xml-id
    C14.S20.XMD7.m1bdec
    0 references
     
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