Complex Analysis

Math 312 Spring 1998  MWF 10:30am - 11:25am
Buckmire  Fowler 112

Class 27 (Wednesday April 1)

SUMMARY Theorems Following Cauchy’s Integral Formulas CURRENT READING Brown & Curchill pages 130-135 NEXT READING Brown & Curchill pages 180-196

Implications of Cauchy’s Integral Formula

We will take some time to see that we understand how the following theorems are derived from the CIF. Cauchy’s Inequality Let f be analytic inside and on a circle |z−z_{0}| = R, taken in the positive sense, and denoted by C_{R}, then the derivatives of f as z_{0} satisfy

|f^{n}(z_{0})| ≤ \frac{n!M}{R^{n}} (n = 1, 2, 3, …)

where |f(z)| ≤ M for all z on C_{R}. \underline{Proof}: (Use the generalized CIF to try and bound f^{(n)}(z_{0}))

Liouville’s Theorem If f is entire and bounded in the complex plane, then f(z) is constant throughout the plane.

\underline{Proof}: (Use Cauchy’s Inequality when n = 1 and assume the radius has to be big enough to include the entire plane, e.g. R → ∞)

Fundamental Theorem of Algebra Every polynomial equation P (z) = a_{0} + a_{1}z + a_{2}z^{2} + ⋯ + a_{n}z^{n} = 0 with degree n ≥ 1 and a_{n} ≠ 0 has atleast one root. \underline{Proof}: The proof is by contradiction and uses Liouville’s Theorem.

Gauss’ mean value theorem If f(z) is analytic inside and on a circle C with center z_{0} and radius r then f(z_{0}) is the mean of the values of f(z) on C, namely

f(z_{0}) = \frac{1}{2π} ∫_{0}^{2π} f(z_{0} + re^{iθ})dθ

\underline{Proof}: This is obvious by using the CIF and a parametrization for |z − z_{0}| = r and interpreting the results.

Maximum modulus theorem If f(z) is analytic inside and on a simple closed curve C and is not identically equal to a constant, then the maximum value of |f(z)| occurs on C. \underline{Proof}:

Minimum modulus theorem If f(z) is analytic inside and on a simple closed curve C and f(z) ≠ 0 inside C, then the minimum value of |f(z)| occurs on C. \underline{Proof}: Use the maximum modulus theorem with g(z) = 1/f(z).

Cauchy’s Residue Theorem If f is analytic on a simple (positively oriented) closed contour C and everywhere inside C except a �nite number of points z_{1}, z_{2}, ⋯ z_{n} inside C, then

∮_{C} f(z) dz = 2πi ∑_{k=1}^{n}Res(z_{k})

Before we talk about what a residue of a function f(z) at z_{k}, denoted by Res(f, z_{k}) or Res(z_{k}) we will need to have a discussion about poles,zeroes and singularities.