DLMF:5.4.E19 (Q2057)

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DLMF:5.4.E19
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    Statements

    Defining formula
    ψ ⁡ ( p q ) = - γ - ln ⁡ q - π 2 ⁢ cot ⁡ ( π ⁢ p q ) + 1 2 ⁢ ∑ k = 1 q - 1 cos ⁡ ( 2 ⁢ π ⁢ k ⁢ p q ) ⁢ ln ⁡ ( 2 - 2 ⁢ cos ⁡ ( 2 ⁢ π ⁢ k q ) ) . digamma 𝑝 𝑞 𝑞 𝜋 2 𝜋 𝑝 𝑞 1 2 superscript subscript 𝑘 1 𝑞 1 2 𝜋 𝑘 𝑝 𝑞 2 2 2 𝜋 𝑘 𝑞 {\displaystyle{\displaystyle\psi\left(\frac{p}{q}\right)=-\gamma-\ln q-\frac{% \pi}{2}\cot\left(\frac{\pi p}{q}\right)+\frac{1}{2}\sum_{k=1}^{q-1}\cos\left(% \frac{2\pi kp}{q}\right)\ln\left(2-2\cos\left(\frac{2\pi k}{q}\right)\right).}}
    0 references
    DLMF id
    DLMF:5.4.E19
    0 references
    Symbols used
    Euler’s constant
    Defining formula
    γ {\displaystyle{\displaystyle\gamma}}
    xml-id
    C5.S2.E3.m2aedec
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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.m2ajdec
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    cosine function
    Defining formula
    cos ⁡ z 𝑧 {\displaystyle{\displaystyle\cos\NVar{z}}}
    xml-id
    C4.S14.E2.m2adec
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    cotangent function
    Defining formula
    cot ⁡ z 𝑧 {\displaystyle{\displaystyle\cot\NVar{z}}}
    xml-id
    C4.S14.E7.m2adec
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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.m2agdec
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    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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    Q11280
    Defining formula
    q 𝑞 {\displaystyle{\displaystyle q}}
    xml-id
    C5.S1.XMD10.m1dec
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    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1adec
    0 references
     
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