DLMF:15.8.E9 (Q5066): Difference between revisions

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Revision as of 14:02, 2 January 2020 Admin (talk | contribs) Bureaucrats, Interface administrators, Administrators 57,118 edits ‎Changed an Item: Add constraint ← Older edit		Latest revision as of 14:02, 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
			Q10755
		Property / Symbols used: Q10755 / rank
			Normal rank
		Property / Symbols used: Q10755 / qualifier
			Defining formula: ${\displaystyle{\displaystyle!}}$ !
		Property / Symbols used: Q10755 / qualifier
			xml-id: introduction.Sx4.p1.t1.r15.m5aadec
		Property / Symbols used
			principal branch of logarithm function
		Property / Symbols used: principal branch of logarithm function / rank
			Normal rank
		Property / Symbols used: principal branch of logarithm function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\ln\NVar{z}}}$ \ln@@{\NVar{z}}
		Property / Symbols used: principal branch of logarithm function / qualifier
			xml-id: C4.S2.E2.m2aadec
		Property / Symbols used
			Q10817
		Property / Symbols used: Q10817 / rank
			Normal rank
		Property / Symbols used: Q10817 / qualifier
			Defining formula: ${\displaystyle{\displaystyle\operatorname{ph}}}$ \phase
		Property / Symbols used: Q10817 / qualifier
			xml-id: C1.S9.E7.m1afdec
		Property / Symbols used
			$$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function
		Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / rank
			Normal rank
		Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
			Defining formula: ${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$ \hyperOlverF@{\NVar{a}}{\NVar{b}}{\NVar{c}}{\NVar{z}}
		Property / Symbols used: $$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function / qualifier
			xml-id: C15.S2.E2.m2afdec
		Property / Symbols used
			Q11644
		Property / Symbols used: Q11644 / rank
			Normal rank
		Property / Symbols used: Q11644 / qualifier
			Defining formula: ${\displaystyle{\displaystyle z}}$ z
		Property / Symbols used: Q11644 / qualifier
			xml-id: C15.S1.XMD3.m1hdec
		Property / Symbols used
			Q11645
		Property / Symbols used: Q11645 / rank
			Normal rank
		Property / Symbols used: Q11645 / qualifier
			Defining formula: ${\displaystyle{\displaystyle a}}$ a
		Property / Symbols used: Q11645 / qualifier
			xml-id: C15.S1.XMD4.m1fdec
		Property / Symbols used
			Q11647
		Property / Symbols used: Q11647 / rank
			Normal rank
		Property / Symbols used: Q11647 / qualifier
			Defining formula: ${\displaystyle{\displaystyle c}}$ c
		Property / Symbols used: Q11647 / qualifier
			xml-id: C15.S1.XMD6.m1hdec
		Property / Symbols used
			Q11658
		Property / Symbols used: Q11658 / rank
			Normal rank
		Property / Symbols used: Q11658 / qualifier
			Defining formula: ${\displaystyle{\displaystyle k}}$ k
		Property / Symbols used: Q11658 / qualifier
			xml-id: C15.S1.XMD7.m1adec
		Property / Symbols used
			Q11651
		Property / Symbols used: Q11651 / rank
			Normal rank
		Property / Symbols used: Q11651 / qualifier
			Defining formula: ${\displaystyle{\displaystyle m}}$ m
		Property / Symbols used: Q11651 / qualifier
			xml-id: C15.S1.XMD9.m1cdec

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Latest revision as of 14:02, 2 January 2020

No description defined

Language	Label	Description	Also known as
English	DLMF:15.8.E9	No description defined

Statements

Defining formula

${\displaystyle{\displaystyle\mathbf{F}\left({a,a+m\atop c};z\right)=\frac{(1-z)^{-a}}{\Gamma\left(a+m\right)\Gamma\left(c-a\right)}\sum_{k=0}^{m-1}\frac{(a)_{k}(c-a-m)_{k}(m-k-1)!}{k!}(z-1)^{-k}+\frac{(-1)^{m}(1-z)^{-a-m}}{\Gamma\left(a\right)\Gamma\left(c-a-m\right)}\sum_{k=0}^{\infty}\frac{(a+m)_{k}(c-a)_{k}}{k!(k+m)!}(1-z)^{-k}\*(\ln\left(1-z\right)+\psi\left(k+1\right)+\psi\left(k+m+1\right)-\psi\left(a+k+m\right)-\psi\left(c-a+k\right)),}}$

0 references

DLMF id

DLMF:15.8.E9

0 references

constraint

${\displaystyle{\displaystyle|z-1|>1,|\operatorname{ph}\left(1-z\right)|<\pi}}$

0 references

Symbols used

gamma function

Defining formula

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

xml-id

C5.S2.E1.m2aedec

0 references

the ratio of the circumference of a circle to its diameter

Defining formula

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

xml-id

C3.S12.E1.m2afdec

0 references

psi (or digamma) function

Defining formula

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

xml-id

C5.S2.E2.m2aadec

0 references

Q10755

Defining formula

${\displaystyle{\displaystyle!}}$

xml-id

introduction.Sx4.p1.t1.r15.m5aadec

0 references

principal branch of logarithm function

Defining formula

${\displaystyle{\displaystyle\ln\NVar{z}}}$

xml-id

C4.S2.E2.m2aadec

0 references

Q10817

Defining formula

${\displaystyle{\displaystyle\operatorname{ph}}}$

xml-id

C1.S9.E7.m1afdec

0 references

$$={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Olver’s hypergeometric function

Defining formula

${\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}$

xml-id

C15.S2.E2.m2afdec

0 references

Q11644

Defining formula

${\displaystyle{\displaystyle z}}$

xml-id

C15.S1.XMD3.m1hdec

0 references

Q11645

Defining formula

${\displaystyle{\displaystyle a}}$

xml-id

C15.S1.XMD4.m1fdec

0 references

Q11647

Defining formula

${\displaystyle{\displaystyle c}}$

xml-id

C15.S1.XMD6.m1hdec

0 references

Q11658

Defining formula

${\displaystyle{\displaystyle k}}$

xml-id

C15.S1.XMD7.m1adec

0 references

Q11651

Defining formula

${\displaystyle{\displaystyle m}}$

xml-id

C15.S1.XMD9.m1cdec

0 references

 

Sitelinks

Wikipedia(0 entries)

DRMF related sites(0 entries)

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