Complex Analysis

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

Class 24 (Wednesday March 25)

SUMMARY Understanding Contour Integration CURRENT READING Brown & Curchill pages 123-125 NEXT READING Brown & Curchill pages 125-129

Update on Class 22 and Class 23 After growing comfortable with evaluating contour integrals using the parametrization method (i.e. using z(t)) we introduced more higher-level tools such as the Cauchy-Goursat Theorem, the Path Independence Theorem and the Deformation Invariance Theorem. \underline{Groupwork} Write down, in your own words, a sentence describing each one of the following theorems. You may also want to write down symbols, pictures or even integrals which help you to understand these theorems. Cauchy-Goursat Theorem

Path Independence Theorem

Deformation Invariance Theorem \underline{Examples} Consider the following integrals of the function f(z) = \frac{1}{z^{3} + z} A = ∮_{C_{1}} f(z) dz, B = ∮_{C_{2}} f(z) dz, C = ∮_{C_{3}} f(z) dz and D = ∮_{C_{4}} f(z) dz The contours C_{1}, C_{2}, C_{3} and C_{4} are as sketched below:

Which of the following equations are true? Give reasons for your answers. 1. A = B?

2. B = C?

3. C = D?

4. D = A? \underline{Exercise} Using partial fractions, we can write \frac{1}{z^{3} + z} = \frac{P}{z + i} + \frac{Q}{z − i} + \frac{R}{z} Find P, Q and R

Evaluate A, B, C and D