Complex Analysis

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

Class 26 (Monday March 30)

SUMMARY Applications and Implications of Cauchy’s Integral Formula CURRENT READING Brown & Curchill pages 123-125 NEXT READING Brown & Curchill pages 125-129

Applications of Cauchy’s Integral Formula

Let C be a simple closed (positively oriented) contour. If f is analytic in some simply connected domain D containing C and z_{0} is any point inside of C, then

∫_{C} \frac{f(z)}{z − z_{0}}dz = 2πif(z_{0})

and

∫_{C} \frac{f(z)}{(z − z_{0})^{m}} dz = \frac{2πi}{(m − 1)!}f^{(m−1)}(z_{0})

These two results lead to a number of other results. Actually, the two formulas are just restatement of one formula, known as the generalized Cauchy Integral Formula. Can you see how the �rst expression (CIF) is just a special case (m =??) of the second one?

\underline{Examples} We have rewritten the integral formulas in the way above so that we can use them to actually evaluate integrals. Let’s to do the following two.

∮_{C} \frac{e^{5z}}{z^{3}} dz =

(where C is |z| = 1 traversed once clockwise)

∫_{C} \frac{2z + 1}{z(z − 1)^{2}} dz =

(where C is given in the sketch)

There are numerous theorems which directly follow from Cauchy’s Integral Formula. I have listed a few of the more famous ones below...

Implications of Cauchy’s Integral Formula

Morera’s Theorem If f(z) is continuous in a simply-connected region R and if ∮_{C} f(z)dz = 0 around every simple closed curve C in R, then f(z) is analytic in R. (NOTE: Morera’s Theorem is the converse of the Cauchy-Goursat theorem.)

Cauchy’s Inequality If f(z) is analytic inside and on a circle of radius r and centered at z = z_{0} then

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

where M is an upper bound on |f(z)| on C

Liouville’s Theorem Suppose that for all z in the entire complex plane, if f(z) is analytic and bounded, (i.e. |f(z)| < M for some real constant M) then f(z) must be a constant.

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.

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

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.

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.

The Argument Theorem Let f(z) be analytic inside and on a simple closed curve C except for a �nite number of poles inside C. Then

\frac{1}{2πi} ∮ \frac{f′(z)}{f(z)}dz = N − P

where N and P are the number of zeroes and poles of f(z) inside C

Rouche’ Theorem If f(z) and g(z) are analytic inside and on a simple closed curve C and if |g(z)| < |f(z)| on C, then f(z) + g(z) and f(z) have the same number of zeros inside of C.