Page 63 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 6.8 Review Questions

1. Suppose f and g are analytic on and inside the simple closed curve C, and suppose moreover
that f(z) = g(z) for all z on C. Prove that f(z) = g(z) for all z inside C.
2. Let C be the ellipse 9x + 4y = 36 traversed once in the counterclockwise direction. Define
2
2
the function g by :
2
s  s 1

g(z)   ds.

C s z
Find : (a) g(i) (b) g(4i)
3. Find :
e 2z
2  dz
C z  4
where, C is the closed curve in the picture:

4. Find 2  e 2z dz, where  is the contour in the picture:
 z  4

5. Evaluate :
sinz
 z 2 dz
C
where, C is a positively oriented closed curve around the origin.

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