DLMF:14.7.E22 (Q4774)

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DLMF:14.7.E22
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    Statements

    Defining formula
    ∑ n = 0 ∞ Q n ⁡ ( x ) ⁢ h n = 1 ( 1 - 2 ⁢ x ⁢ h + h 2 ) 1 / 2 ⁢ ln ⁡ ( x - h + ( 1 - 2 ⁢ x ⁢ h + h 2 ) 1 / 2 ( x 2 - 1 ) 1 / 2 ) . superscript subscript 𝑛 0 shorthand-Legendre-Q-second-kind 𝑛 𝑥 superscript ℎ 𝑛 1 superscript 1 2 𝑥 ℎ superscript ℎ 2 1 2 𝑥 ℎ superscript 1 2 𝑥 ℎ superscript ℎ 2 1 2 superscript superscript 𝑥 2 1 1 2 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}Q_{n}\left(x\right)h^{n}=\frac{% 1}{\left(1-2xh+h^{2}\right)^{1/2}}\*\ln\left(\frac{x-h+\left(1-2xh+h^{2}\right% )^{1/2}}{\left(x^{2}-1\right)^{1/2}}\right).}}
    0 references
    DLMF id
    DLMF:14.7.E22
    0 references
    Symbols used
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2acdec
    0 references
    Q11448
    Defining formula
    Q ν ⁡ ( z ) = Q ν 0 ⁡ ( z ) shorthand-Legendre-Q-second-kind 𝜈 𝑧 Legendre-Q-second-kind 0 𝜈 𝑧 {\displaystyle{\displaystyle Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}% \left(z\right)}}
    xml-id
    C14.S2.SS2.p2.m5acdec
    0 references
    Q11589
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C14.S1.XMD1.m1udec
    0 references
    Q11602
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C14.S1.XMD6.m1sdec
    0 references
     
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