DLMF:4.23.E20 (Q1772)

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DLMF:4.23.E20
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    Statements

    Defining formula
    arcsin ⁡ x = 1 2 ⁢ π ± i ⁢ ln ⁡ ( ( x 2 - 1 ) 1 / 2 + x ) , 𝑥 plus-or-minus 1 2 𝜋 𝑖 superscript superscript 𝑥 2 1 1 2 𝑥 {\displaystyle{\displaystyle\operatorname{arcsin}x=\tfrac{1}{2}\pi\pm i\ln% \left((x^{2}-1)^{1/2}+x\right),}}
    0 references
    DLMF id
    DLMF:4.23.E20
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    constraint
    x ∈ [ 1 , ∞ ) 𝑥 1 {\displaystyle{\displaystyle x\in[1,\infty)}}
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    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aedec
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    Q11243
    Defining formula
    [ a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p2.t1.r1.m4adec
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    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2aadec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2aedec
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    Q11242
    Defining formula
    arcsin ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m5aedec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
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