14: Analytic Continuation and the Gamma Function
- Page ID
- 6563
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In this topic we will look at the Gamma function. This is an important and fascinating function that generalizes factorials from integers to all complex numbers. We look at a few of its many interesting properties. In particular, we will look at its connection to the Laplace transform. We will start by discussing the notion of analytic continuation. We will see that we have, in fact, been using this already without any comment. This was a little sloppy mathematically speaking and we will make it more precise here.
- 14.1: Analytic Continuation
- If we have an function which is analytic on a region A, we can sometimes extend the function to be analytic on a bigger region. This is called analytic continuation.
Thumbnail: Analytic continuation from \(U\) (centered at 1) to \(V\) (centered at a=(3+i)/2). (CC BY-SA 4.0 International; Ncsinger via Wikipedia)