DLMF:4.38.E2 (Q1952): Difference between revisions

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Property / constraint
 

ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}

\Re z>0
Property / constraint: ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} / rank
 
Normal rank
Property / constraint
 

| z | > 1 𝑧 1 {\displaystyle{\displaystyle|z|>1}}

|z|>1
Property / constraint: | z | > 1 𝑧 1 {\displaystyle{\displaystyle|z|>1}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q11247 / rank
 
Normal rank
Property / Symbols used: Q11247 / qualifier
 
test:

arcsinh ⁡ z hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}

\asinh@@{\NVar{z}}
Property / Symbols used: Q11247 / qualifier
 
xml-id: C4.S37.SS2.p1.m8aadec
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.m2adec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
test:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1adec
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.m1adec

Latest revision as of 18:19, 1 January 2020

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English
DLMF:4.38.E2
No description defined

    Statements

    test
    arcsinh ⁡ z = ln ⁡ ( 2 ⁢ z ) + 1 2 ⁢ 1 2 ⁢ z 2 - 1 ⋅ 3 2 ⋅ 4 ⁢ 1 4 ⁢ z 4 + 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 ⁢ 1 6 ⁢ z 6 - ⋯ , hyperbolic-inverse-sine 𝑧 2 𝑧 1 2 1 2 superscript 𝑧 2 ⋅ 1 3 ⋅ 2 4 1 4 superscript 𝑧 4 ⋅ 1 3 5 ⋅ 2 4 6 1 6 superscript 𝑧 6 ⋯ {\displaystyle{\displaystyle\operatorname{arcsinh}z=\ln\left(2z\right)+\frac{1% }{2}\frac{1}{2z^{2}}-\frac{1\cdot 3}{2\cdot 4}\frac{1}{4z^{4}}+\frac{1\cdot 3% \cdot 5}{2\cdot 4\cdot 6}\frac{1}{6z^{6}}-\cdots,}}
    0 references
    DLMF id
    DLMF:4.38.E2
    0 references
    constraint
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
    0 references
    ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
    0 references
    | z | > 1 𝑧 1 {\displaystyle{\displaystyle|z|>1}}
    0 references
    Symbols used
    Q11247
    test
    arcsinh ⁡ z hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m8aadec
    0 references
    principal branch of logarithm function
    test
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
    0 references
    Q11217
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C4.S1.XMD8.m1adec
    0 references
     
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