DLMF:5.11.E8 (Q2130): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10931 / rank
 
Normal rank
Property / Symbols used: Q10931 / qualifier
 
Defining formula:

∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}

\asympexp
Property / Symbols used: Q10931 / qualifier
 
xml-id: C2.S1.SS3.p1.m11acdec
Property / Symbols used
 
Property / Symbols used: the ratio of the circumference of a circle to its diameter / rank
 
Normal rank
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
Defining formula:

π {\displaystyle{\displaystyle\pi}}

\cpi
Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
 
xml-id: C3.S12.E1.m2acdec
Property / Symbols used
 
Property / Symbols used: Q10815 / rank
 
Normal rank
Property / Symbols used: Q10815 / qualifier
 
Defining formula:

[ a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b}]}}

[\NVar{a},\NVar{b}]
Property / Symbols used: Q10815 / qualifier
 
xml-id: introduction.Sx4.p1.t1.r30.m6adec
Property / Symbols used
 
Property / Symbols used: general logarithm function / rank
 
Normal rank
Property / Symbols used: general logarithm function / qualifier
 
Defining formula:

Ln ⁡ z multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\operatorname{Ln}\NVar{z}}}

\Ln@@{\NVar{z}}
Property / Symbols used: general logarithm function / qualifier
 
xml-id: C4.S2.E1.m2aadec
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.m2abdec
Property / Symbols used
 
Property / Symbols used: Q11279 / rank
 
Normal rank
Property / Symbols used: Q11279 / qualifier
 
Defining formula:

k 𝑘 {\displaystyle{\displaystyle k}}

k
Property / Symbols used: Q11279 / qualifier
 
xml-id: C5.S1.XMD4.m1edec
Property / Symbols used
 
Property / Symbols used: Q11273 / rank
 
Normal rank
Property / Symbols used: Q11273 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11273 / qualifier
 
xml-id: C5.S1.XMD7.m1ddec

Latest revision as of 19:41, 1 January 2020

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DLMF:5.11.E8
No description defined

    Statements

    Defining formula
    Ln ⁡ Γ ⁡ ( z + h ) ∼ ( z + h - 1 2 ) ⁢ ln ⁡ z - z + 1 2 ⁢ ln ⁡ ( 2 ⁢ π ) + ∑ k = 2 ∞ ( - 1 ) k ⁢ B k ⁡ ( h ) k ⁢ ( k - 1 ) ⁢ z k - 1 , asymptotic-expansion multivalued-natural-logarithm Euler-Gamma 𝑧 ℎ 𝑧 ℎ 1 2 𝑧 𝑧 1 2 2 𝜋 superscript subscript 𝑘 2 superscript 1 𝑘 Bernoulli-polynomial-B 𝑘 ℎ 𝑘 𝑘 1 superscript 𝑧 𝑘 1 {\displaystyle{\displaystyle\operatorname{Ln}\Gamma\left(z+h\right)\sim\left(z% +h-\tfrac{1}{2}\right)\ln z-z+\tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=2}^{% \infty}\frac{(-1)^{k}B_{k}\left(h\right)}{k(k-1)z^{k-1}},}}
    0 references
    DLMF id
    DLMF:5.11.E8
    0 references
    Symbols used
    Q11288
    Defining formula
    B n ⁡ ( x ) Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{\NVar{n}}\left(\NVar{x}\right)}}
    xml-id
    C24.S2.SS1.m2adec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Q10931
    Defining formula
    ∼ asymptotic-expansion {\displaystyle{\displaystyle\sim}}
    xml-id
    C2.S1.SS3.p1.m11acdec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    Q10815
    Defining formula
    [ a , b ] 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b}]}}
    xml-id
    introduction.Sx4.p1.t1.r30.m6adec
    0 references
    general logarithm function
    Defining formula
    Ln ⁡ z multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\operatorname{Ln}\NVar{z}}}
    xml-id
    C4.S2.E1.m2aadec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2abdec
    0 references
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1edec
    0 references
    Q11273
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1ddec
    0 references
     
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