DLMF:5.4.E18 (Q2056)

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DLMF:5.4.E18
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    Statements

    test
    ℑ ⁡ ψ ⁡ ( 1 + i ⁢ y ) = - 1 2 ⁢ y + π 2 ⁢ coth ⁡ ( π ⁢ y ) . digamma 1 𝑖 𝑦 1 2 𝑦 𝜋 2 hyperbolic-cotangent 𝜋 𝑦 {\displaystyle{\displaystyle\Im\psi\left(1+iy\right)=-\frac{1}{2y}+\frac{\pi}{% 2}\coth\left(\pi y\right).}}
    0 references
    DLMF id
    DLMF:5.4.E18
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    test
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aidec
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    psi (or digamma) function
    test
    ψ ⁡ ( z ) digamma 𝑧 {\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E2.m2afdec
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    hyperbolic cotangent function
    test
    coth ⁡ z hyperbolic-cotangent 𝑧 {\displaystyle{\displaystyle\coth\NVar{z}}}
    xml-id
    C4.S28.E7.m2aadec
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    Q10901
    test
    ℑ ⁡ absent {\displaystyle{\displaystyle\Im}}
    xml-id
    C1.S9.E2.m2abdec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
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    Q11278
    test
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C5.S1.XMD6.m1edec
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