Complex Analysis

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

Class 25 (Friday March 27)

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

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

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

The CIF leads to some of the most astonishing results in complex analysis. It is a truly amazing idea; that the value of an analytic function at a point z_{0} in a simply-connected domain depends on values it takes on some closed contour C encircling the point.

An alternative proof of the result is reasonably straightforward and involves the continuity of f(z) at every point in D and the formula for bounding a contour integral. You might try reading it on page 123-124 of the text.

Higher Derivatives of Analytic Functions Here is the �rst of many amazing ideas derived from the CIF. 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

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

and in fact you should be able to write down a general formula for the n^{th} derivative of f(z) evaluated at z_{0} in terms of a contour integral:

f^{(n)}(z_{0}) =

This is an amazing result, because it means that when a function is analytic then all of its higher derivatives exist and are also each analytic! \underline{Exercise}

∫_{|2z|=3} \frac{e^{z^{2}}}{(z + i)(z + 2)} dz = ∫_{|z|=3} \frac{e^{z^{2}}}{(z + i)(z + 2)} dz =

\underline{Groupwork}

1.∮_{C} \frac{x^{2} − y^{2}}{2} + xyi dz  C : |z − i| = 2 counter-clockwise

2.∮_{C} \overline{z} dz,  C : |z| = 2 clockwise

3.∮_{C} \frac{dz}{z^{2} + π^{2}}  C : |z| = 3 counter-clockwise

4.∮_{C} \frac{sinh(2z)}{z^{2} + π^{2}} dz  C : |z + i| = 3 counter-clockwise

5.∮_{C} \frac{dz}{(z − 3)^{4}}  C : |z − 2| = 2 twice counter-clockwise

\underline{Exercise} Evaluate the following integral ∮_{C} \frac{z + i}{z^{3} + 2z^{2}}dz where the contour C is (a) the circle |z| = 1 traversed once counter clockwise

(b) the circle |z + 2 − i| = 2 traversed once counter clockwise

(c) the circle |z − 2i| = 1 traversed once counter clockwise