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DLMF:4.38.E7 - MaRDI portal

DLMF:4.38.E7 (Q2395)

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DLMF:4.38.E7
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    Statements

    DLMF defining formula
    arctanh ⁡ z = z 1 - z 2 ⁢ ( 1 + 2 3 ⁢ z 2 z 2 - 1 + 2 ⋅ 4 3 ⋅ 5 ⁢ ( z 2 z 2 - 1 ) 2 + ⋯ ) , hyperbolic-inverse-tangent 𝑧 𝑧 1 superscript 𝑧 2 1 2 3 superscript 𝑧 2 superscript 𝑧 2 1 ⋅ 2 4 ⋅ 3 5 superscript superscript 𝑧 2 superscript 𝑧 2 1 2 ⋯ {\displaystyle{\displaystyle\operatorname{arctanh}z=\frac{z}{1-z^{2}}\*{\left(% 1+\frac{2}{3}\frac{z^{2}}{z^{2}-1}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}% {z^{2}-1}\right)^{2}+\cdots\right)},}}
    0 references
    constraint
    ℜ ⁡ ( z 2 ) < 1 2 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\Re(z^{2})<\tfrac{1}{2}}}
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    Symbols used
    Q11219
    DLMF defining formula
    arctanh ⁡ z hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m10abdec
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    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1abdec
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    complex variable
    DLMF defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C4.S1.XMD8.m1fdec
    0 references
     
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