holomorphic function associated with continuous function
Proof. The right hand side of (2) is defined since its integrand is continuous. On has to show that it equals
Let , . We may write first
As well, we have a positive constant such that
When we choose , it follows that
and, by the estimating theorem of contour integral,
where is the length of the contour. The last expression tends to zero as . This settles the proof.
Remark 1. By induction, one can prove the following generalisation of (2):
Remark 2. The contour may be . If it especially is a circle, then (1) defines a holomorphic function inside and another outside it.
|Title||holomorphic function associated with continuous function|
|Date of creation||2013-03-22 19:14:29|
|Last modified on||2013-03-22 19:14:29|
|Last modified by||pahio (2872)|