DLMF:4.23.E23 (Q1775)

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DLMF:4.23.E23
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    Statements

    Defining formula
    arccos ⁡ z = - 2 ⁢ i ⁢ ln ⁡ ( ( 1 + z 2 ) 1 / 2 + i ⁢ ( 1 - z 2 ) 1 / 2 ) , 𝑧 2 𝑖 superscript 1 𝑧 2 1 2 𝑖 superscript 1 𝑧 2 1 2 {\displaystyle{\displaystyle\operatorname{arccos}z=-2i\ln\left(\left(\frac{1+z% }{2}\right)^{1/2}+i\left(\frac{1-z}{2}\right)^{1/2}\right),}}
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    DLMF id
    DLMF:4.23.E23
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    constraint
    z ∈ ℂ ∖ ( - ∞ , - 1 ) ∪ ( 1 , ∞ ) 𝑧 1 1 {\displaystyle{\displaystyle z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)}}
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    Symbols used
    Q10923
    Defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2abdec
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    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2addec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
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    Q11208
    Defining formula
    arccos ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m6aedec
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    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
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    Q10777
    Defining formula
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6abdec
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    Q10785
    Defining formula
    ∖ {\displaystyle{\displaystyle\setminus}}
    xml-id
    introduction.Sx4.p2.t1.r19.m2abdec
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    Q10824
    Defining formula
    ∪ {\displaystyle{\displaystyle\cup}}
    xml-id
    introduction.Sx4.p1.t1.r28.m2abdec
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