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Incomplete Gamma Functions
| Content Provider | Scilit |
|---|---|
| Author | Olver, Frank |
| Copyright Year | 1997 |
| Description | If Rea 2 1 , then by uniform convergence we may expand e-' in ascending powers o f t and integrate term by term. In this way we obtain the following expansion, valid for all z : This enables y (a ,z) to be continued analytically with respect to a into the left halfplane, or with respect to z outside the principal phase range. Thus it is seen that when z # 0 the only singularities of y(a,z) as a function of a are simple poles at a = 0, - 1, -2, .. .. Also, if a is fixed, then the branch of y (a, z ) obtained aRer z encircles the origin m times is given by y (a, zeZmxi) = elmrmi y (a, Z ) (a # 0, - 1 , - 2, . . .). (5.03) 5.2 The complementary incomplete Gamma function, or Prym's function as it is sometimes called, is defined by r ( a , z ) = e-t ta-l dt, there being no restriction on a. The principal branch is defined in the same way as for y (a, z). Combination with (5.01) yields y (a, Z ) + r ( a , I ) = r (a ) . (5.05) From (5.03) and (5.05) we derive Analytic continuation shows that this result also holds when a is zero or a negative integer, provided that the right-hand side is replaced by its limiting value. Ex. 5.1 In the notation of 543 and 4, show that Book Name: Asymptotics and Special Functions |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2010-0-47263-4&isbn=9780429064616&doi=10.1201/9781439864548-22&format=pdf |
| Ending Page | 63 |
| Page Count | 1 |
| Starting Page | 63 |
| DOI | 10.1201/9781439864548-22 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 1997-01-24 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Asymptotics and Special Functions Function Convergence Notation Singularities Analytic Continuation Respect Integer Halfplane Encircles |
| Content Type | Text |
| Resource Type | Chapter |
