DLMF:4.38.E4 (Q1954): Difference between revisions

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Property / constraint
 

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

\realpart@@{z}>0
Property / constraint: ℜ ⁡ z > 0 𝑧 0 {\displaystyle{\displaystyle\Re z>0}} / rank
 
Normal rank
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 | ≤ 2 𝑧 1 2 {\displaystyle{\displaystyle|z-1|\leq 2}}

|z-1|\leq 2
Property / constraint: | z - 1 | ≤ 2 𝑧 1 2 {\displaystyle{\displaystyle|z-1|\leq 2}} / rank
 
Normal rank
Property / Symbols used
 
Property / Symbols used: Q10755 / rank
 
Normal rank
Property / Symbols used: Q10755 / qualifier
 
test:

! {\displaystyle{\displaystyle!}}

!
Property / Symbols used: Q10755 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r15.m5adec
Property / Symbols used
 
Property / Symbols used: Q11246 / rank
 
Normal rank
Property / Symbols used: Q11246 / qualifier
 
test:

arccosh ⁡ z hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}

\acosh@@{\NVar{z}}
Property / Symbols used: Q11246 / qualifier
 
xml-id: C4.S37.SS2.p1.m9aadec
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.m1aadec
Property / Symbols used
 
Property / Symbols used: Q11223 / rank
 
Normal rank
Property / Symbols used: Q11223 / qualifier
 
test:

n 𝑛 {\displaystyle{\displaystyle n}}

n
Property / Symbols used: Q11223 / qualifier
 
xml-id: C4.S1.XMD3.m1dec
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.m1cdec

Latest revision as of 18:19, 1 January 2020

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English
DLMF:4.38.E4
No description defined

    Statements

    test
    arccosh ⁡ z = ( 2 ⁢ ( z - 1 ) ) 1 / 2 ⁢ ( 1 + ∑ n = 1 ∞ ( - 1 ) n ⁢ 1 ⋅ 3 ⋅ 5 ⁢ ⋯ ⁢ ( 2 ⁢ n - 1 ) 2 2 ⁢ n ⁢ n ! ⁢ ( 2 ⁢ n + 1 ) ⁢ ( z - 1 ) n ) , hyperbolic-inverse-cosine 𝑧 superscript 2 𝑧 1 1 2 1 superscript subscript 𝑛 1 superscript 1 𝑛 ⋅ 1 3 5 ⋯ 2 𝑛 1 superscript 2 2 𝑛 𝑛 2 𝑛 1 superscript 𝑧 1 𝑛 {\displaystyle{\displaystyle\operatorname{arccosh}z=(2(z-1))^{1/2}\*{\left(1+% \sum_{n=1}^{\infty}(-1)^{n}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}n!(2n+1)}(% z-1)^{n}\right)},}}
    0 references
    DLMF id
    DLMF:4.38.E4
    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 | ≤ 2 𝑧 1 2 {\displaystyle{\displaystyle|z-1|\leq 2}}
    0 references
    Symbols used
    Q10755
    test
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5adec
    0 references
    Q11246
    test
    arccosh ⁡ z hyperbolic-inverse-cosine 𝑧 {\displaystyle{\displaystyle\operatorname{arccosh}\NVar{z}}}
    xml-id
    C4.S37.SS2.p1.m9aadec
    0 references
    Q10811
    test
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
    0 references
    Q11223
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C4.S1.XMD3.m1dec
    0 references
    Q11217
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C4.S1.XMD8.m1cdec
    0 references
     
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