DLMF:4.23.E36 (Q1788)

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DLMF:4.23.E36
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    Statements

    Defining formula
    arctan ⁡ z = 1 2 ⁢ arctan ⁡ ( 2 ⁢ x 1 - x 2 - y 2 ) + 1 4 ⁢ i ⁢ ln ⁡ ( x 2 + ( y + 1 ) 2 x 2 + ( y - 1 ) 2 ) , 𝑧 1 2 2 𝑥 1 superscript 𝑥 2 superscript 𝑦 2 1 4 𝑖 superscript 𝑥 2 superscript 𝑦 1 2 superscript 𝑥 2 superscript 𝑦 1 2 {\displaystyle{\displaystyle\operatorname{arctan}z=\tfrac{1}{2}\operatorname{% arctan}\left(\frac{2x}{1-x^{2}-y^{2}}\right)+\tfrac{1}{4}i\ln\left(\frac{x^{2}% +(y+1)^{2}}{x^{2}+(y-1)^{2}}\right),}}
    0 references
    DLMF id
    DLMF:4.23.E36
    0 references
    Symbols used
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2apdec
    0 references
    Q10816
    Defining formula
    arctan ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arctan}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m7agdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2akdec
    0 references
    Q11218
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C4.S1.XMD6.m1fdec
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    Q11222
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C4.S1.XMD7.m1cdec
    0 references
    Q11217
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C4.S1.XMD8.m1aedec
    0 references
     
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