DLMF:4.37.E25 (Q1944)

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DLMF:4.37.E25
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    Statements

    test
    arctanh ⁡ x = ± 1 2 ⁢ π ⁢ i + 1 2 ⁢ ln ⁡ ( x + 1 x - 1 ) , hyperbolic-inverse-tangent 𝑥 plus-or-minus 1 2 𝜋 𝑖 1 2 𝑥 1 𝑥 1 {\displaystyle{\displaystyle\operatorname{arctanh}x=\pm\tfrac{1}{2}\pi i+% \tfrac{1}{2}\ln\left(\frac{x+1}{x-1}\right),}}
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    DLMF id
    DLMF:4.37.E25
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    constraint
    x ∈ ( - ∞ , - 1 ) ∪ ( 1 , ∞ ) 𝑥 1 1 {\displaystyle{\displaystyle x\in(-\infty,-1)\cup(1,\infty)}}
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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.m2afdec
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    Q10784
    test
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2ahdec
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    Q11248
    test
    arctanh ⁡ z hyperbolic-inverse-tangent 𝑧 {\displaystyle{\displaystyle\operatorname{arctanh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m10addec
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    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
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    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aidec
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    Q10777
    test
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6acdec
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    Q10824
    test
    ∪ {\displaystyle{\displaystyle\cup}}
    xml-id
    introduction.Sx4.p1.t1.r28.m2abdec
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    Q11218
    test
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C4.S1.XMD6.m1bdec
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