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--- math121a-f23:september_20_wednesday [2023/09/21 12:16]
pzhou
+++ math121a-f23:september_20_wednesday [2023/09/21 12:30] (current)
pzhou [Exercises]
@@ Line 35: / Line 35: @@
 ===== Method 3: change of variable =====
 Let  $w = 1/z$ , then we have
-I = \oint_{|w|=1/10, CW} \frac{1}{(w^{-1}-1)(w^{-1}-2)} d (w^{-1}) = \oint_{|w|=1/10, CW} \frac{-1}{(1-w)(1-2w)} dw = \oint_{|w|=1/10} \frac{1}{(1-w)(1-2w)}} dw
-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.
+ I = \oint_{|w|=1/10, CW} \frac{1}{(w^{-1}-1)(w^{-1}-2)} d (w^{-1}) = \oint_{|w|=1/10, CW} \frac{-1}{(1-w)(1-2w)} dw = \oint_{|w|=1/10} \frac{1}{(1-w)(1-2w)} dw
+where in the last step, I changed the orientation of the contour from  $CW$  to  $CCW$ (CCW is by default, hence omited) and add an extra  $(-1)$  factor to the integral.
+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)

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