DLMF:25.11.E35 (Q7709): Difference between revisions

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Revision as of 13:45, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 13:45, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint
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		Property / Symbols used
			Q12161
		Property / Symbols used: Q12161 / rank
			Normal rank
		Property / Symbols used: Q12161 / qualifier
			Defining formula: ${\displaystyle{\displaystyle a}}$ a
		Property / Symbols used: Q12161 / qualifier
			xml-id: C25.S1.XMD6.m1acdec
		Property / Symbols used
			Q12149
		Property / Symbols used: Q12149 / rank
			Normal rank
		Property / Symbols used: Q12149 / qualifier
			Defining formula: ${\displaystyle{\displaystyle s}}$ s
		Property / Symbols used: Q12149 / qualifier
			xml-id: C25.S1.XMD7.m1xdec

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Latest revision as of 13:45, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:25.11.E35	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x=2^{-s}\left(\zeta\left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right),}}$

0 references

DLMF id

DLMF:25.11.E35

0 references

constraint

${\displaystyle{\displaystyle\Re a>0}}$

0 references

${\displaystyle{\displaystyle\Re s>0}}$

0 references

${\displaystyle{\displaystyle\Re a=0}}$

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${\displaystyle{\displaystyle\Im a\neq 0}}$

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${\displaystyle{\displaystyle 0<\Re s<1}}$

0 references

Symbols used

gamma function

Defining formula

${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$

xml-id

C5.S2.E1.m2aidec

0 references

Hurwitz zeta function

Defining formula

${\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}$

xml-id

C25.S11.E1.m2aaidec

0 references

Q10770

Defining formula

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

xml-id

C1.S4.SS4.m1andec

0 references

base of natural logarithm

Defining formula

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

xml-id

C4.S2.E11.m2afdec

0 references

Q10901

Defining formula

${\displaystyle{\displaystyle\Im}}$

xml-id

C1.S9.E2.m2adec

0 references

Q10771

Defining formula

${\displaystyle{\displaystyle\int}}$

xml-id

C1.S4.SS4.m3andec

0 references

Q10811

Defining formula

${\displaystyle{\displaystyle\Re}}$

xml-id

C1.S9.E2.m1ardec

0 references

Q12148

Defining formula

${\displaystyle{\displaystyle n}}$

xml-id

C25.S1.XMD3.m1rdec

0 references

Q12152

Defining formula

${\displaystyle{\displaystyle x}}$

xml-id

C25.S1.XMD5.m1odec

0 references

Q12161

Defining formula

${\displaystyle{\displaystyle a}}$

xml-id

C25.S1.XMD6.m1acdec

0 references

Q12149

Defining formula

${\displaystyle{\displaystyle s}}$

xml-id

C25.S1.XMD7.m1xdec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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