PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] Barnes' integral representation of the hypergeometric function (Theorem)

When $ a,b,c,d$ are complex numbers and $ z$ is a complex number such that $ -\pi < \arg (-z) < +\pi$ and $ C$ is a contour in the complex $ s$-plane which goes from $ -i \infty$ to $ + i \infty$ chosen such that the poles of $ \Gamma (a+s) \Gamma (b+s)$ lie to the left of $ C$ and the poles of $ \Gamma (-s)$ lie to the right of $ C$, then

$\displaystyle \int_C {\Gamma (a+s) \Gamma (b+s) \over \Gamma (c+s)} \Gamma (-s) (-z)^s \, ds = 2 \pi i {\Gamma (a) \Gamma (b) \over \Gamma (c)} F (a, b; c; z) $



"Barnes' integral representation of the hypergeometric function" is owned by rspuzio.
(view preamble)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: right, poles, contour, complex numbers

This is version 1 of Barnes' integral representation of the hypergeometric function, born on 2007-10-28.
Object id is 10019, canonical name is BarnesIntegralRepresentationOfTheHypergeometricFunction.
Accessed 396 times total.

Classification:
AMS MSC: 33C05 (Special functions :: Hypergeometric functions :: Classical hypergeometric functions, $_2F_1$)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)