DLMF:4.37.E17 (Q1936): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
Defining formula:

i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
Defining formula:

ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}

\ln@@{\NVar{z}}
Property / Symbols used: principal branch of logarithm function / qualifier
 
xml-id: C4.S2.E2.m2aadec
Property / Symbols used
 
Property / Symbols used: Q11222 / rank
 
Normal rank
Property / Symbols used: Q11222 / qualifier
 
Defining formula:

y 𝑦 {\displaystyle{\displaystyle y}}

y
Property / Symbols used: Q11222 / qualifier
 
xml-id: C4.S1.XMD7.m1dec

Latest revision as of 19:16, 1 January 2020

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DLMF:4.37.E17
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    Statements

    Defining formula
    arcsinh ⁡ ( i ⁢ y ) = 1 2 ⁢ π ⁢ i ± ln ⁡ ( ( y 2 - 1 ) 1 / 2 + y ) , hyperbolic-inverse-sine 𝑖 𝑦 plus-or-minus 1 2 𝜋 𝑖 superscript superscript 𝑦 2 1 1 2 𝑦 {\displaystyle{\displaystyle\operatorname{arcsinh}\left(iy\right)=\tfrac{1}{2}% \pi i\pm\ln\left((y^{2}-1)^{1/2}+y\right),}}
    0 references
    DLMF id
    DLMF:4.37.E17
    0 references
    constraint
    y ∈ [ 1 , ∞ ) 𝑦 1 {\displaystyle{\displaystyle y\in[1,\infty)}}
    0 references
    Symbols used
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q11243
    Defining formula
    [ a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p2.t1.r1.m4adec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2aadec
    0 references
    Q11247
    Defining formula
    arcsinh ⁡ z hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m8addec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2aadec
    0 references
    Q11222
    Defining formula
    y 𝑦 {\displaystyle{\displaystyle y}}
    xml-id
    C4.S1.XMD7.m1dec
    0 references
     
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