DLMF:32.2.E6 (Q9159)

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DLMF:32.2.E6
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    Statements

    Defining formula
    d 2 w d z 2 = 1 2 ⁢ ( 1 w + 1 w - 1 + 1 w - z ) ⁢ ( d w d z ) 2 - ( 1 z + 1 z - 1 + 1 w - z ) ⁢ d w d z + w ⁢ ( w - 1 ) ⁢ ( w - z ) z 2 ⁢ ( z - 1 ) 2 ⁢ ( α + β ⁢ z w 2 + γ ⁢ ( z - 1 ) ( w - 1 ) 2 + δ ⁢ z ⁢ ( z - 1 ) ( w - z ) 2 ) , derivative 𝑤 𝑧 2 1 2 1 𝑤 1 𝑤 1 1 𝑤 𝑧 superscript derivative 𝑤 𝑧 2 1 𝑧 1 𝑧 1 1 𝑤 𝑧 derivative 𝑤 𝑧 𝑤 𝑤 1 𝑤 𝑧 superscript 𝑧 2 superscript 𝑧 1 2 𝛼 𝛽 𝑧 superscript 𝑤 2 𝛾 𝑧 1 superscript 𝑤 1 2 𝛿 𝑧 𝑧 1 superscript 𝑤 𝑧 2 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=\frac{% 1}{2}\left(\frac{1}{w}+\frac{1}{w-1}+\frac{1}{w-z}\right)\left(\frac{\mathrm{d% }w}{\mathrm{d}z}\right)^{2}-\left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{w-z}% \right)\frac{\mathrm{d}w}{\mathrm{d}z}+\frac{w(w-1)(w-z)}{z^{2}(z-1)^{2}}\left% (\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1)^{2}}+\frac{\delta z(z-1% )}{(w-z)^{2}}\right),}}
    0 references
    DLMF id
    DLMF:32.2.E6
    0 references
    Symbols used
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2aedec
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    Q12523
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C32.S1.XMD4.m1edec
    0 references
    Q12524
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C32.S2.XMD1.m1ddec
    0 references
    Q12525
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C32.S2.XMD2.m1cdec
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    Q12526
    Defining formula
    γ 𝛾 {\displaystyle{\displaystyle\gamma}}
    xml-id
    C32.S2.XMD3.m1bdec
    0 references
    Q12527
    Defining formula
    δ 𝛿 {\displaystyle{\displaystyle\delta}}
    xml-id
    C32.S2.XMD4.m1bdec
    0 references
     
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