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--- math121a-f23:september_20_wednesday [2023/09/21 12:18]
pzhou [Method 3: change of variable]
+++ math121a-f23:september_20_wednesday [2023/09/21 12:30] (current)
pzhou [Exercises]
@@ Line 41: / Line 41: @@
 Since the integrand is only singular at  $w=1,1/2$ , and the contour  $|w|=1/10$  contains no singularity in its interior, the integral is 0.
+===== Riemann sphere =====
+It is useful to think of add a point  $\infty$  to the complex plane  $\C$ , and think of  $\C \cup \{\infty\}$  as a sphere, where  $\infty$  is identified with the north pole,  $0$  with the south pole, the unit circle  $|z|=1$  as the equator.
+The natural coordinate to use near the north pole is  $w=1/z$ , so that  $z=\infty$  corresponds to  $w=0$ .
+===== Exercises =====
+Let  $C$  be the contour of  $|z|=10$ . Consider the following integrals.
+(1)  $\oint_C \frac{1}{1+z^2} dz$
+(2) (the result for this one is not zero.)
+ $\oint_C \frac{z}{1+z^2} dz$
+(3)  $\oint_C \frac{z^2}{1+z^4} dz$
+Apply methods 1,2,3 to the above problems (each method need to be used once)

Lecture Notes