DLMF:4.33.E3 (Q1860)

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DLMF:4.33.E3
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    Statements

    Defining formula
    tanh ⁡ z = z - z 3 3 + 2 15 ⁢ z 5 - 17 315 ⁢ z 7 + ⋯ + 2 2 ⁢ n ⁢ ( 2 2 ⁢ n - 1 ) ⁢ B 2 ⁢ n ( 2 ⁢ n ) ! ⁢ z 2 ⁢ n - 1 + ⋯ , 𝑧 𝑧 superscript 𝑧 3 3 2 15 superscript 𝑧 5 17 315 superscript 𝑧 7 ⋯ superscript 2 2 𝑛 superscript 2 2 𝑛 1 Bernoulli-number-B 2 𝑛 2 𝑛 superscript 𝑧 2 𝑛 1 ⋯ {\displaystyle{\displaystyle\tanh z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{% 17}{315}z^{7}+\cdots+\frac{2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,}}
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    DLMF id
    DLMF:4.33.E3
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    constraint
    | z | < 1 2 ⁢ π 𝑧 1 2 𝜋 {\displaystyle{\displaystyle|z|<\frac{1}{2}\pi}}
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    Symbols used
    Q11109
    Defining formula
    B n Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{\NVar{n}}}}
    xml-id
    C24.S2.SS1.m1adec
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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.m2adec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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