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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Write a walkthrough for the proof of Theorem 4.24 (Cauchy’s
at least three figures and elaborate on the proof given, filling in the gaps you find.
![Theorem 4.24. If f is holomorphic in an open set containing D[w, R] then
f(w)
f(2)
- dz -
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5958936e-64d8-4f9f-ac16-af104f1eb32d%2Ffa89dc57-525c-418f-b029-ec80e9624414%2Fl0sz6j2_processed.png&w=3840&q=75)
Transcribed Image Text:Theorem 4.24. If f is holomorphic in an open set containing D[w, R] then
f(w)
f(2)
- dz -
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.
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