DLMF:5.17.E4 (Q2184): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q10770 / rank
 
Normal rank
Property / Symbols used: Q10770 / qualifier
 
Defining formula:

d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}

\diff{\NVar{x}}
Property / Symbols used: Q10770 / qualifier
 
xml-id: C1.S4.SS4.m1adec
Property / Symbols used
 
Property / Symbols used: Q10771 / rank
 
Normal rank
Property / Symbols used: Q10771 / qualifier
 
Defining formula:

∫ {\displaystyle{\displaystyle\int}}

\int
Property / Symbols used: Q10771 / qualifier
 
xml-id: C1.S4.SS4.m3adec
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.m2adec
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.m2adec
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.m1bdec

Latest revision as of 01:09, 2 January 2020

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DLMF:5.17.E4
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    Statements

    Defining formula
    Ln ⁡ G ⁡ ( z + 1 ) = 1 2 ⁢ z ⁢ ln ⁡ ( 2 ⁢ π ) - 1 2 ⁢ z ⁢ ( z + 1 ) + z ⁢ Ln ⁡ Γ ⁡ ( z + 1 ) - ∫ 0 z Ln ⁡ Γ ⁡ ( t + 1 ) ⁢ d t . multivalued-natural-logarithm Barnes-Gamma 𝑧 1 1 2 𝑧 2 𝜋 1 2 𝑧 𝑧 1 𝑧 multivalued-natural-logarithm Euler-Gamma 𝑧 1 superscript subscript 0 𝑧 multivalued-natural-logarithm Euler-Gamma 𝑡 1 𝑡 {\displaystyle{\displaystyle\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z% \ln\left(2\pi\right)-\tfrac{1}{2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1% \right)-\int_{0}^{z}\operatorname{Ln}\Gamma\left(t+1\right)\mathrm{d}t.}}
    0 references
    DLMF id
    DLMF:5.17.E4
    0 references
    Symbols used
    Q11295
    Defining formula
    G ⁡ ( z ) Barnes-Gamma 𝑧 {\displaystyle{\displaystyle G\left(\NVar{z}\right)}}
    xml-id
    C5.S17.E1.m1acdec
    0 references
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2aadec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    general logarithm function
    Defining formula
    Ln ⁡ z multivalued-natural-logarithm 𝑧 {\displaystyle{\displaystyle\operatorname{Ln}\NVar{z}}}
    xml-id
    C4.S2.E1.m2adec
    0 references
    principal branch of logarithm function
    Defining formula
    ln ⁡ z 𝑧 {\displaystyle{\displaystyle\ln\NVar{z}}}
    xml-id
    C4.S2.E2.m2adec
    0 references
    Q11273
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C5.S1.XMD7.m1bdec
    0 references
     
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