Theorem 4.24. If f is holomorphic in an open set containing D[w,R] then f(w) = Ze dz. 2ni Jc»R] z – w This is Cauchy's Integral Formula for the case that the integration path is a circle; we will prove the general statement at the end of this chapter. However, already this special case is worth meditating over: the data on the right-hand side of Theorem 4.24 is entirely given by the values that f(z) takes on for z on the circle C[w, R]. Thus Cauchy's Integral Formula says that this data determines f(w). This has the flavor of mean-value theorems, which the following corollary makes even more apparent.

Theorem 4.24. If f is holomorphic in an open set containing D[w,R] then f(w) = Ze dz. 2ni Jc»R] z – w This is Cauchy's Integral Formula for the case that the integration path is a circle; we will prove the general statement at the end of this chapter. However, already this special case is worth meditating over: the data on the right-hand side of Theorem 4.24 is entirely given by the values that f(z) takes on for z on the circle C[w, R]. Thus Cauchy's Integral Formula says that this data determines f(w). This has the flavor of mean-value theorems, which the following corollary makes even more apparent.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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