DLMF:25.11.E25 (Q7699): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Hurwitz zeta function / rank
 
Normal rank
Property / Symbols used: Hurwitz zeta function / qualifier
 
Defining formula:

ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}

\Hurwitzzeta@{\NVar{s}}{\NVar{a}}
Property / Symbols used: Hurwitz zeta function / qualifier
 
xml-id: C25.S11.E1.m2azdec
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.m1aedec
Property / Symbols used
 
Property / Symbols used: base of natural logarithm / rank
 
Normal rank
Property / Symbols used: base of natural logarithm / qualifier
 
Defining formula:

e {\displaystyle{\displaystyle\mathrm{e}}}

\expe
Property / Symbols used: base of natural logarithm / qualifier
 
xml-id: C4.S2.E11.m2adec
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.m3aedec
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

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

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aidec
Property / Symbols used
 
Property / Symbols used: Q12152 / rank
 
Normal rank
Property / Symbols used: Q12152 / qualifier
 
Defining formula:

x 𝑥 {\displaystyle{\displaystyle x}}

x
Property / Symbols used: Q12152 / qualifier
 
xml-id: C25.S1.XMD5.m1gdec
Property / Symbols used
 
Property / Symbols used: Q12161 / rank
 
Normal rank
Property / Symbols used: Q12161 / qualifier
 
Defining formula:

a 𝑎 {\displaystyle{\displaystyle a}}

a
Property / Symbols used: Q12161 / qualifier
 
xml-id: C25.S1.XMD6.m1tdec
Property / Symbols used
 
Property / Symbols used: Q12149 / rank
 
Normal rank
Property / Symbols used: Q12149 / qualifier
 
Defining formula:

s 𝑠 {\displaystyle{\displaystyle s}}

s
Property / Symbols used: Q12149 / qualifier
 
xml-id: C25.S1.XMD7.m1qdec

Latest revision as of 13:41, 2 January 2020

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English
DLMF:25.11.E25
No description defined

    Statements

    Defining formula
    ζ ⁡ ( s , a ) = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 1 - e - x ⁢ d x , Hurwitz-zeta 𝑠 𝑎 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 1 superscript 𝑒 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{\Gamma\left(s\right% )}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-e^{-x}}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.11.E25
    0 references
    constraint
    ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}}
    0 references
    ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}}
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2addec
    0 references
    Hurwitz zeta function
    Defining formula
    ζ ⁡ ( s , a ) Hurwitz-zeta 𝑠 𝑎 {\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}
    xml-id
    C25.S11.E1.m2azdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1aedec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3aedec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aidec
    0 references
    Q12152
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C25.S1.XMD5.m1gdec
    0 references
    Q12161
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C25.S1.XMD6.m1tdec
    0 references
    Q12149
    Defining formula
    s 𝑠 {\displaystyle{\displaystyle s}}
    xml-id
    C25.S1.XMD7.m1qdec
    0 references
     
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