DLMF:25.11.E32 (Q7706): Difference between revisions

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Revision as of 23:03, 29 December 2019 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Created a new Item: Wikidata Toolkit example test item creation		Latest revision as of 13:44, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint
(15 intermediate revisions by the same user not shown)
		Property / constraint
			${\displaystyle{\displaystyle\Re a>0}}$ \realpart@@{a}>0
		Property / constraint: ${\displaystyle{\displaystyle\Re a>0}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle n=1,2,\dots}}$ n=1,2,\dots
		Property / constraint: ${\displaystyle{\displaystyle n=1,2,\dots}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle\Re a>0}}$ \Re a>0
		Property / constraint: ${\displaystyle{\displaystyle\Re a>0}}$ / rank
			Normal rank
		Property / Symbols used
			Q11109
		Property / Symbols used: Q11109 / rank
			Normal rank
		Property / Symbols used: Q11109 / qualifier
			Defining formula: ${\displaystyle{\displaystyle B_{\NVar{n}}}}$ \BernoullinumberB{\NVar{n}}
		Property / Symbols used: Q11109 / qualifier
			xml-id: C24.S2.SS1.m1ahdec
		Property / Symbols used
			Hurwitz zeta function
		Property / Symbols used: Hurwitz zeta function / rank
			Normal rank
		Property / Symbols used: Hurwitz zeta function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\zeta\left(\NVar{s},\NVar{a}\right)}}$ \Hurwitzzeta@{\NVar{s}}{\NVar{a}}
		Property / Symbols used: Hurwitz zeta function / qualifier
			xml-id: C25.S11.E1.m2aagdec
		Property / Symbols used
			Riemann zeta function
		Property / Symbols used: Riemann zeta function / rank
			Normal rank
		Property / Symbols used: Riemann zeta function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}$ \Riemannzeta@{\NVar{s}}
		Property / Symbols used: Riemann zeta function / qualifier
			xml-id: C25.S2.E1.m2agdec
		Property / Symbols used
			Q10754
		Property / Symbols used: Q10754 / rank
			Normal rank
		Property / Symbols used: Q10754 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}$ \binom{\NVar{m}}{\NVar{n}}
		Property / Symbols used: Q10754 / qualifier
			xml-id: C1.S2.SS1.m1aadec
		Property / Symbols used
			Q10770
		Property / Symbols used: Q10770 / rank
			Normal rank
		Property / Symbols used: Q10770 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$ \diff{\NVar{x}}
		Property / Symbols used: Q10770 / qualifier
			xml-id: C1.S4.SS4.m1aldec
		Property / Symbols used
			psi (or digamma) function
		Property / Symbols used: psi (or digamma) function / rank
			Normal rank
		Property / Symbols used: psi (or digamma) function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$ \digamma@{\NVar{z}}
		Property / Symbols used: psi (or digamma) function / qualifier
			xml-id: C5.S2.E2.m2acdec
		Property / Symbols used
			Q10771
		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.m3aldec
		Property / Symbols used
			Q10811
		Property / Symbols used: Q10811 / rank
			Normal rank
		Property / Symbols used: Q10811 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\Re}}$ \realpart@@
		Property / Symbols used: Q10811 / qualifier
			xml-id: C1.S9.E2.m1apdec
		Property / Symbols used
			Q12150
		Property / Symbols used: Q12150 / rank
			Normal rank
		Property / Symbols used: Q12150 / qualifier
			Defining formula: ${\displaystyle{\displaystyle k}}$ k
		Property / Symbols used: Q12150 / qualifier
			xml-id: C25.S1.XMD1.m1ddec
		Property / Symbols used
			Q12148
		Property / Symbols used: Q12148 / rank
			Normal rank
		Property / Symbols used: Q12148 / qualifier
			Defining formula: ${\displaystyle{\displaystyle n}}$ n
		Property / Symbols used: Q12148 / qualifier
			xml-id: C25.S1.XMD3.m1odec
		Property / Symbols used
			Q12152
		Property / Symbols used: Q12152 / rank
			Normal rank
		Property / Symbols used: Q12152 / qualifier
			Defining formula: ${\displaystyle{\displaystyle x}}$ x
		Property / Symbols used: Q12152 / qualifier
			xml-id: C25.S1.XMD5.m1mdec
		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.m1aadec
		Property / Symbols used
			Q12164
		Property / Symbols used: Q12164 / rank
			Normal rank
		Property / Symbols used: Q12164 / qualifier
			Defining formula: ${\displaystyle{\displaystyle h(n)}}$ h(n)
		Property / Symbols used: Q12164 / qualifier
			xml-id: C25.S11.XMD3.m1dec

---

Latest revision as of 13:44, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:25.11.E32	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n\right)+(-1)^{n}h(n)\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}h(k)\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},}}$

0 references

DLMF id

DLMF:25.11.E32

0 references

constraint

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

0 references

${\displaystyle{\displaystyle n=1,2,\dots}}$

0 references

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

0 references

Symbols used

Q11109

Defining formula

${\displaystyle{\displaystyle B_{\NVar{n}}}}$

xml-id

C24.S2.SS1.m1ahdec

0 references

Hurwitz zeta function

Defining formula

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

xml-id

C25.S11.E1.m2aagdec

0 references

Riemann zeta function

Defining formula

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

xml-id

C25.S2.E1.m2agdec

0 references

Q10754

Defining formula

${\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}}}$

xml-id

C1.S2.SS1.m1aadec

0 references

Q10770

Defining formula

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

xml-id

C1.S4.SS4.m1aldec

0 references

psi (or digamma) function

Defining formula

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

xml-id

C5.S2.E2.m2acdec

0 references

Q10771

Defining formula

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

xml-id

C1.S4.SS4.m3aldec

0 references

Q10811

Defining formula

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

xml-id

C1.S9.E2.m1apdec

0 references

Q12150

Defining formula

${\displaystyle{\displaystyle k}}$

xml-id

C25.S1.XMD1.m1ddec

0 references

Q12148

Defining formula

${\displaystyle{\displaystyle n}}$

xml-id

C25.S1.XMD3.m1odec

0 references

Q12152

Defining formula

${\displaystyle{\displaystyle x}}$

xml-id

C25.S1.XMD5.m1mdec

0 references

Q12161

Defining formula

${\displaystyle{\displaystyle a}}$

xml-id

C25.S1.XMD6.m1aadec

0 references

Q12164

Defining formula

${\displaystyle{\displaystyle h(n)}}$

xml-id

C25.S11.XMD3.m1dec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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