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--- math121a-f23:september_20_wednesday [2023/09/21 19:12]
pzhou created
+++ math121a-f23:september_20_wednesday [2026/02/21 14:41] (current)
@@ Line 29: / Line 29: @@
 on the contour $|z|=R$. Thus
 $$ \oint_{|z|=R} \frac{1}{|z-1| |z-2| }  |dz| \leq \frac{1}{(R-1)(R-2)}  \oint |dz| = \frac{1}{(R-1)(R-2)}  (2\pi R) $$
-To summarize, we have
+To summarize, we have $I_R$ is a constant, and for any $R>10$, we have
-$$ 0 \leq |I_R| \leq \frac{2\pi R}{(R-1)(R-2)}. $$
+$$ 0 \leq |I_R| \leq \frac{2\pi R}{(R-1)(R-2)}.  $$
+Thus $|I_R|=0$, hence $I_R=0$.
+===== Method 3: change of variable =====
+Let $w = 1/z$, then we have
+$$
+ I = \oint_{|w|=1/10, CW} \frac{1}{(w^{-1}-1)(w^{-1}-2)} d (w^{-1}) = \oint_{|w|=1/10, CW} \frac{-1}{(1-w)(1-2w)} dw = \oint_{|w|=1/10} \frac{1}{(1-w)(1-2w)} dw $$
+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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