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

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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:

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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DLMF:4.38.E2
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    Statements

    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
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    z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
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    z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}}
    0 references
    | z | > 1 𝑧 1 {\displaystyle{\displaystyle|z|>1}}
    0 references
    arcsinh z hyperbolic-inverse-sine 𝑧 {\displaystyle{\displaystyle\operatorname{arcsinh}\NVar{z}}}
    C4.S37.SS2.p1.m8aadec
    0 references
    ln z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    C4.S2.E2.m2adec
    0 references
    absent {\displaystyle{\displaystyle\Re}}
    C1.S9.E2.m1adec
    0 references
    z 𝑧 {\displaystyle{\displaystyle z}}
    C4.S1.XMD8.m1adec
    0 references