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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods

Fall 2024 • Question 2

Question

Math 6720 • Applied Complex Variables and Asymptotic Methods Fall 2024 • Question 2

(20 pts) Prove the Cauchy Integral Formula: Let \(f\) be analytic in an open domain \(D\), and let \(z\in D\). Then for any non-negative integer \(n\), \[ f^{(n)}(z)=\frac {n!}{2\pi i}\oint _C \frac {f(w)}{(w-z)^{n+1}}\,dw, \] where \(C\) is any simple contour in \(D\) enclosing \(z\). You may take the \(n=0,1\) versions of this formula as given.

Raw LaTeX

(20 pts) Prove the Cauchy Integral Formula: Let \(f\) be analytic in an open domain \(D\), and let
\(z\in D\). Then for any non-negative integer \(n\),
\[
f^{(n)}(z)=\frac{n!}{2\pi i}\oint_C \frac{f(w)}{(w-z)^{n+1}}\,dw,
\]
where \(C\) is any simple contour in \(D\) enclosing \(z\). You may take the \(n=0,1\) versions of this
formula as given.

Topic Tags

Cauchy Integral Formula and Integral Theorems Contour Integration and Residue Calculus Harmonic Functions and Poisson Integral Formula Keyhole and Sector Contours