DLMF:14.12.E3 (Q4824): Difference between revisions

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Revision as of 15:39, 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 01:42, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint
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		Property / constraint
			${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}$ \realpart@@{\mu}>-\tfrac{1}{2}
		Property / constraint: ${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}$ \realpart@@{(\nu\pm\mu)}>-1
		Property / constraint: ${\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle 0<\theta<\pi}}$ 0<\theta<\pi
		Property / constraint: ${\displaystyle{\displaystyle 0<\theta<\pi}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}$ \Re\mu>-\tfrac{1}{2}
		Property / constraint: ${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}$ / rank
			Normal rank
		Property / constraint
			${\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}$ \Re(\nu\pm\mu)>-1
		Property / constraint: ${\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}$ / rank
			Normal rank
		Property / Symbols used
			gamma function
		Property / Symbols used: gamma function / rank
			Normal rank
		Property / Symbols used: gamma function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$ \EulerGamma@{\NVar{z}}
		Property / Symbols used: gamma function / qualifier
			xml-id: C5.S2.E1.m2abdec
		Property / Symbols used
			Ferrers function of the second kind
		Property / Symbols used: Ferrers function of the second kind / rank
			Normal rank
		Property / Symbols used: Ferrers function of the second kind / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathsf{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)}}$ \FerrersQ[\NVar{\mu}]{\NVar{\nu}}@{\NVar{x}}
		Property / Symbols used: Ferrers function of the second kind / qualifier
			xml-id: C14.S3.E2.m2adec
		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.m2aadec
		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.m2aadec
		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.m1abdec
		Property / Symbols used
			hyperbolic cosine function
		Property / Symbols used: hyperbolic cosine function / rank
			Normal rank
		Property / Symbols used: hyperbolic cosine function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\cosh\NVar{z}}}$ \cosh@@{\NVar{z}}
		Property / Symbols used: hyperbolic cosine function / qualifier
			xml-id: C4.S28.E2.m2adec
		Property / Symbols used
			hyperbolic sine function
		Property / Symbols used: hyperbolic sine function / rank
			Normal rank
		Property / Symbols used: hyperbolic sine function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\sinh\NVar{z}}}$ \sinh@@{\NVar{z}}
		Property / Symbols used: hyperbolic sine function / qualifier
			xml-id: C4.S28.E1.m2adec
		Property / Symbols used
			imaginary unit
		Property / Symbols used: imaginary unit / rank
			Normal rank
		Property / Symbols used: imaginary unit / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathrm{i}}}$ \iunit
		Property / Symbols used: imaginary unit / qualifier
			xml-id: C1.S9.E1.m2adec
		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.m3abdec
		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.m1abdec
		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.m2aadec
		Property / Symbols used
			Q11591
		Property / Symbols used: Q11591 / rank
			Normal rank
		Property / Symbols used: Q11591 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mu}}$ \mu
		Property / Symbols used: Q11591 / qualifier
			xml-id: C14.S1.XMD7.m1bdec
		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.m1bdec

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Latest revision as of 01:42, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:14.12.E3	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{\pi^{1/2}\Gamma\left(\nu+\mu+1\right)(\sin\theta)^{\mu}}{2^{\mu+1}\Gamma\left(\mu+\frac{1}{2}\right)\Gamma\left(\nu-\mu+1\right)}\*\left(\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta+i\sin\theta\cosh t)^{\nu+\mu+1}}\mathrm{d}t+\int_{0}^{\infty}\frac{(\sinh t)^{2\mu}}{(\cos\theta-i\sin\theta\cosh t)^{\nu+\mu+1}}\mathrm{d}t\right),}}$

0 references

DLMF id

DLMF:14.12.E3

0 references

constraint

${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}$

0 references

${\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}$

0 references

${\displaystyle{\displaystyle 0<\theta<\pi}}$

0 references

${\displaystyle{\displaystyle\Re\mu>-\tfrac{1}{2}}}$

0 references

${\displaystyle{\displaystyle\Re(\nu\pm\mu)>-1}}$

0 references

Symbols used

gamma function

Defining formula

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

xml-id

C5.S2.E1.m2abdec

0 references

Ferrers function of the second kind

Defining formula

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

xml-id

C14.S3.E2.m2adec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

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

xml-id

C3.S12.E1.m2aadec

0 references

cosine function

Defining formula

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

xml-id

C4.S14.E2.m2aadec

0 references

Q10770

Defining formula

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

xml-id

C1.S4.SS4.m1abdec

0 references

hyperbolic cosine function

Defining formula

${\displaystyle{\displaystyle\cosh\NVar{z}}}$

xml-id

C4.S28.E2.m2adec

0 references

hyperbolic sine function

Defining formula

${\displaystyle{\displaystyle\sinh\NVar{z}}}$

xml-id

C4.S28.E1.m2adec

0 references

imaginary unit

Defining formula

${\displaystyle{\displaystyle\mathrm{i}}}$

xml-id

C1.S9.E1.m2adec

0 references

Q10771

Defining formula

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

xml-id

C1.S4.SS4.m3abdec

0 references

Q10811

Defining formula

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

xml-id

C1.S9.E2.m1abdec

0 references

sine function

Defining formula

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

xml-id

C4.S14.E1.m2aadec

0 references

Q11591

Defining formula

${\displaystyle{\displaystyle\mu}}$

xml-id

C14.S1.XMD7.m1bdec

0 references

Q11590

Defining formula

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

xml-id

C14.S1.XMD8.m1bdec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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