DLMF:4.23.E23 (Q1775): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: imaginary unit / rank
 
Normal rank
Property / Symbols used: imaginary unit / qualifier
 
test:

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

\iunit
Property / Symbols used: imaginary unit / qualifier
 
xml-id: C1.S9.E1.m2ahdec
Property / Symbols used
 
Property / Symbols used: Q11208 / rank
 
Normal rank
Property / Symbols used: Q11208 / qualifier
 
test:

arccos ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\NVar{z}}}

\acos@@{\NVar{z}}
Property / Symbols used: Q11208 / qualifier
 
xml-id: C4.S23.SS2.p1.m6aedec
Property / Symbols used
 
Property / Symbols used: principal branch of logarithm function / rank
 
Normal rank
Property / Symbols used: principal branch of logarithm function / qualifier
 
test:

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.m2addec
Property / Symbols used
 
Property / Symbols used: Q10777 / rank
 
Normal rank
Property / Symbols used: Q10777 / qualifier
 
test:

( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}

(\NVar{a},\NVar{b})
Property / Symbols used: Q10777 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r29.m6abdec
Property / Symbols used
 
Property / Symbols used: Q10785 / rank
 
Normal rank
Property / Symbols used: Q10785 / qualifier
 
test:

∖ {\displaystyle{\displaystyle\setminus}}

\setminus
Property / Symbols used: Q10785 / qualifier
 
xml-id: introduction.Sx4.p2.t1.r19.m2abdec
Property / Symbols used
 
Property / Symbols used: Q10824 / rank
 
Normal rank
Property / Symbols used: Q10824 / qualifier
 
test:

∪ {\displaystyle{\displaystyle\cup}}

\cup
Property / Symbols used: Q10824 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r28.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11217 / rank
 
Normal rank
Property / Symbols used: Q11217 / qualifier
 
test:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11217 / qualifier
 
xml-id: C4.S1.XMD8.m1udec

Latest revision as of 17:33, 1 January 2020

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English
DLMF:4.23.E23
No description defined

    Statements

    test
    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),}}
    0 references
    DLMF id
    DLMF:4.23.E23
    0 references
    constraint
    z ∈ ℂ ∖ ( - ∞ , - 1 ) ∪ ( 1 , ∞ ) 𝑧 1 1 {\displaystyle{\displaystyle z\in\mathbb{C}\setminus(-\infty,-1)\cup(1,\infty)}}
    0 references
    Symbols used
    Q10923
    test
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2abdec
    0 references
    Q10784
    test
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2addec
    0 references
    imaginary unit
    test
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2ahdec
    0 references
    Q11208
    test
    arccos ⁡ z 𝑧 {\displaystyle{\displaystyle\operatorname{arccos}\NVar{z}}}
    xml-id
    C4.S23.SS2.p1.m6aedec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2addec
    0 references
    Q10777
    test
    ( a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle(\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p1.t1.r29.m6abdec
    0 references
    Q10785
    test
    ∖ {\displaystyle{\displaystyle\setminus}}
    xml-id
    introduction.Sx4.p2.t1.r19.m2abdec
    0 references
    Q10824
    test
    ∪ {\displaystyle{\displaystyle\cup}}
    xml-id
    introduction.Sx4.p1.t1.r28.m2abdec
    0 references
    Q11217
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C4.S1.XMD8.m1udec
    0 references
     
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