DLMF:4.23.E34 (Q1786)

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DLMF:4.23.E34
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    Statements

    Defining formula
    arcsin ⁡ z = arcsin ⁡ β + i ⁢ sign ⁡ ( y ) ⁢ ln ⁡ ( α + ( α 2 - 1 ) 1 / 2 ) , 𝑧 𝛽 imaginary-unit sign 𝑦 𝛼 superscript superscript 𝛼 2 1 1 2 {\displaystyle{\displaystyle\operatorname{arcsin}z=\operatorname{arcsin}\beta+% \mathrm{i}\operatorname{sign}\left(y\right)\ln\left(\alpha+(\alpha^{2}-1)^{1/2% }\right),}}
    0 references
    DLMF id
    DLMF:4.23.E34
    0 references
    Symbols used
    Q10815
    Defining formula
    [ a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b}]}}
    xml-id
    introduction.Sx4.p1.t1.r30.m6adec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2aidec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2andec
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    Q11242
    Defining formula
    arcsin ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arcsin}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m5ahdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aidec
    0 references
    Q11245
    Defining formula
    ∉ {\displaystyle{\displaystyle\notin}}
    xml-id
    introduction.Sx4.p1.t1.r11.m2adec
    0 references
    Q10777
    Defining formula
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6addec
    0 references
    Q10897
    Defining formula
    sign ⁡ x sign 𝑥 {\displaystyle{\displaystyle\operatorname{sign}\NVar{x}}}
    xml-id
    introduction.Sx4.p2.t1.r18.m6adec
    0 references
    Q11218
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C4.S1.XMD6.m1ddec
    0 references
    Q11222
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C4.S1.XMD7.m1adec
    0 references
    Q11217
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C4.S1.XMD8.m1acdec
    0 references
    DLMF:4.23.E37
    Defining formula
    α 𝛼 {\displaystyle{\displaystyle\alpha}}
    xml-id
    C4.S23.XMD2.m1dec
    0 references
    DLMF:4.23.E38
    Defining formula
    β 𝛽 {\displaystyle{\displaystyle\beta}}
    xml-id
    C4.S23.XMD3.m1dec
    0 references
     
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