DLMF:14.17.E14 (Q4895): Difference between revisions

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Revision as of 17:53, 30 December 2019 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 13:25, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint
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		Property / constraint
			${\displaystyle{\displaystyle\Re\nu>0}}$ \realpart@@{\nu}>0
		Property / constraint: ${\displaystyle{\displaystyle\Re\nu>0}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle\Re\lambda>0}}$ \Re\lambda>0
		Property / constraint: ${\displaystyle{\displaystyle\Re\lambda>0}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle\Re\nu>0}}$ \Re\nu>0
		Property / constraint: ${\displaystyle{\displaystyle\Re\nu>0}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle\lambda\neq\nu}}$ \lambda\neq\nu
		Property / constraint: ${\displaystyle{\displaystyle\lambda\neq\nu}}$ / rank
			Normal rank
		Property / Symbols used
			the ratio of the circumference of a circle to its diameter
		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: ${\displaystyle{\displaystyle\pi}}$ \cpi
		Property / Symbols used: the ratio of the circumference of a circle to its diameter / qualifier
			xml-id: C3.S12.E1.m2addec
		Property / Symbols used
			cosine function
		Property / Symbols used: cosine function / rank
			Normal rank
		Property / Symbols used: cosine function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\cos\NVar{z}}}$ \cos@@{\NVar{z}}
		Property / Symbols used: cosine function / qualifier
			xml-id: C4.S14.E2.m2abdec
		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.m1amdec
		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.m2addec
		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.m3amdec
		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.m1aadec
		Property / Symbols used
			Q11595
		Property / Symbols used: Q11595 / rank
			Normal rank
		Property / Symbols used: Q11595 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)}}$ \FerrersP[]{\NVar{\nu}}@{\NVar{x}}=\FerrersP[0]{\nu}@{x}
		Property / Symbols used: Q11595 / qualifier
			xml-id: C14.S2.SS2.p2.m2abdec
		Property / Symbols used
			Q11596
		Property / Symbols used: Q11596 / rank
			Normal rank
		Property / Symbols used: Q11596 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)}}$ \FerrersQ[]{\NVar{\nu}}@{\NVar{x}}=\FerrersQ[0]{\nu}@{x}
		Property / Symbols used: Q11596 / qualifier
			xml-id: C14.S2.SS2.p2.m4abdec
		Property / Symbols used
			sine function
		Property / Symbols used: sine function / rank
			Normal rank
		Property / Symbols used: sine function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\sin\NVar{z}}}$ \sin@@{\NVar{z}}
		Property / Symbols used: sine function / qualifier
			xml-id: C4.S14.E1.m2acdec
		Property / Symbols used
			Q11589
		Property / Symbols used: Q11589 / rank
			Normal rank
		Property / Symbols used: Q11589 / qualifier
			Defining formula: ${\displaystyle{\displaystyle x}}$ x
		Property / Symbols used: Q11589 / qualifier
			xml-id: C14.S1.XMD1.m1mdec
		Property / Symbols used
			Q11590
		Property / Symbols used: Q11590 / rank
			Normal rank
		Property / Symbols used: Q11590 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\nu}}$ \nu
		Property / Symbols used: Q11590 / qualifier
			xml-id: C14.S1.XMD8.m1idec

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Latest revision as of 13:25, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:14.17.E14	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle\int_{-1}^{1}\mathsf{P}_{\nu}\left(x\right)\mathsf{Q}_{\lambda}\left(x\right)\mathrm{d}x=\frac{2\sin\left(\nu\pi\right)\cos\left(\lambda\pi\right)\left(\psi\left(\nu+1\right)-\psi\left(\lambda+1\right)\right)+\pi\cos\left((\lambda-\nu)\pi\right)-\pi}{\pi(\lambda-\nu)(\lambda+\nu+1)},}}$

0 references

DLMF id

DLMF:14.17.E14

0 references

constraint

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

0 references

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

0 references

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

0 references

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

0 references

${\displaystyle{\displaystyle\lambda\neq\nu}}$

0 references

Symbols used

the ratio of the circumference of a circle to its diameter

Defining formula

${\displaystyle{\displaystyle\pi}}$

xml-id

C3.S12.E1.m2addec

0 references

cosine function

Defining formula

${\displaystyle{\displaystyle\cos\NVar{z}}}$

xml-id

C4.S14.E2.m2abdec

0 references

Q10770

Defining formula

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

xml-id

C1.S4.SS4.m1amdec

0 references

psi (or digamma) function

Defining formula

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

xml-id

C5.S2.E2.m2addec

0 references

Q10771

Defining formula

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

xml-id

C1.S4.SS4.m3amdec

0 references

Q10811

Defining formula

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

xml-id

C1.S9.E2.m1aadec

0 references

Q11595

Defining formula

${\displaystyle{\displaystyle\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)}}$

xml-id

C14.S2.SS2.p2.m2abdec

0 references

Q11596

Defining formula

${\displaystyle{\displaystyle\mathsf{Q}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{Q}^{0}_{\nu}\left(x\right)}}$

xml-id

C14.S2.SS2.p2.m4abdec

0 references

sine function

Defining formula

${\displaystyle{\displaystyle\sin\NVar{z}}}$

xml-id

C4.S14.E1.m2acdec

0 references

Q11589

Defining formula

${\displaystyle{\displaystyle x}}$

xml-id

C14.S1.XMD1.m1mdec

0 references

Q11590

Defining formula

${\displaystyle{\displaystyle\nu}}$

xml-id

C14.S1.XMD8.m1idec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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