DLMF:14.11.E5 (Q4821)

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DLMF:14.11.E5
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    Statements

    Defining formula
    ∂ ∂ ⁡ μ ⁡ 𝖰 ν μ ⁡ ( x ) | μ = 0 = - 1 4 ⁢ π 2 ⁢ 𝖯 ν ⁡ ( x ) + ( ψ ⁡ ( - ν ) - π ⁢ cot ⁡ ( ν ⁢ π ) ) ⁢ 𝖰 ν ⁡ ( x ) . evaluated-at partial-derivative 𝜇 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 𝜇 0 1 4 superscript 𝜋 2 shorthand-Ferrers-Legendre-P-first-kind 𝜈 𝑥 digamma 𝜈 𝜋 𝜈 𝜋 shorthand-Ferrers-Legendre-Q-first-kind 𝜈 𝑥 {\displaystyle{\displaystyle\left.\frac{\partial}{\partial\mu}\mathsf{Q}^{\mu}% _{\nu}\left(x\right)\right|_{\mu=0}=-\tfrac{1}{4}\pi^{2}\mathsf{P}_{\nu}\left(% x\right)+\left(\psi\left(-\nu\right)-\pi\cot\left(\nu\pi\right)\right)\mathsf{% Q}_{\nu}\left(x\right).}}
    0 references
    DLMF id
    DLMF:14.11.E5
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    Symbols used
    Ferrers function of the second kind
    Defining formula
    𝖰 ν μ ⁡ ( x ) Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝑥 {\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}% \right)}}
    xml-id
    C14.S3.E2.m2aadec
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    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2addec
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    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2acdec
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    psi (or digamma) function
    Defining formula
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2abdec
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