update notations and fix typos

pages/Math416/Math416_L10.md

@@ -34,7 +34,7 @@ $$
 \left|\sum_{n=1}^{K}a_n-\sum_{n=1}^{L}a_{\sigma(n)}\right|<2\epsilon
 $$
 
-EOP
+QED
 
 ## Chapter 4 Complex Integration
 
@@ -72,7 +72,7 @@ $$
 
 Assume $\phi$ is continuous on $[a,b]$, the equality means $\lambda(t)\phi(t)$ is real and positive everywhere on $[a,b]$, which means $\arg \phi(t)$ is constant.
 
-EOP
+QED
 
 #### Definition 6.4 Arc Length
 
@@ -82,12 +82,12 @@ $$
 \Gamma=\int_a^b |\gamma'(t)| dt=\int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2} dt
 $$
 
-N.B. If $\int_{\Gamma} f(\zeta) d\zeta$ depends on orientation of $\Gamma$, but not the parametrization.
+N.B. If $\int_{\Gamma} f(z) dz$ depends on orientation of $\Gamma$, but not the parametrization.
 
 We define
 
 $$
-\int_{\Gamma} f(\zeta) d\zeta=\int_{\Gamma} f(\gamma(t))\gamma'(t) dt
+\int_{\Gamma} f(z) dz=\int_{\Gamma} f(\gamma(t))\gamma'(t) dt
 $$
 
 Example:
@@ -95,7 +95,7 @@ Example:
 Suppose $\Gamma$ is the circle centered at $z_0$ with radius $R$
 
 $$
-\int_{\Gamma} \frac{1}{\zeta-z_0} d\zeta
+\int_{\Gamma} \frac{1}{z-z_0} dz
 $$
 
 Parameterize the unit circle:
@@ -106,7 +106,7 @@ $$
 $$
 
 $$
-f(\zeta)=\frac{1}{\zeta-z_0}
+f(z)=\frac{1}{z-z_0}
 $$
 
 $$
@@ -114,7 +114,7 @@ f(\gamma(t))=\frac{1}{(z_0+Re^{it})-z_0}
 $$
 
 $$
-\int_{\Gamma} f(\zeta) d\zeta=\int_0^{2\pi} f(\gamma(t))\gamma'(t) dt=\int_0^{2\pi} \frac{1}{Re^{-it}}iRe^{it} dt=2\pi i
+\int_{\Gamma} f(z) dz=\int_0^{2\pi} f(\gamma(t))\gamma'(t) dt=\int_0^{2\pi} \frac{1}{Re^{-it}}iRe^{it} dt=2\pi i
 $$
 
 #### Theorem 6.11 (Uniform Convergence)
@@ -142,7 +142,7 @@ $$
 \end{aligned}
 $$
 
-EOP
+QED
 
 #### Theorem 6.6 (Integral of derivative)
 
@@ -157,20 +157,20 @@ $$
 \end{aligned}
 $$
 
-EOP
+QED
 
 Example:
 
 Let $R$ be a rectangle $\{-a,a,ai+b,ai-b\}$, $\Gamma$ is the boundary of $R$ with positive orientation.
 
-Let $\int_{R} e^{-\zeta^2}d\zeta$.
+Let $\int_{R} e^{-z^2}dz$.
 
-Is $e^{-\zeta^2}=\frac{d}{d\zeta}f(\zeta)$?
+Is $e^{-z^2}=\frac{d}{dz}f(z)$?
 
 Yes, since
 
 $$
-e^{\zeta^2}=1-\frac{\zeta^2}{1!}+\frac{\zeta^4}{2!}-\frac{\zeta^6}{3!}+\cdots=\frac{d}{d\zeta}\left(\frac{\zeta}{1!}-\frac{1}{3}\frac{\zeta^3}{2!}+\frac{1}{5}\frac{\zeta^5}{3!}-\cdots\right)
+e^{z^2}=1-\frac{z^2}{1!}+\frac{z^4}{2!}-\frac{z^6}{3!}+\cdots=\frac{d}{dz}\left(\frac{z}{1!}-\frac{1}{3}\frac{z^3}{2!}+\frac{1}{5}\frac{z^5}{3!}-\cdots\right)
 $$
 
 This is polynomial, therefore holomorphic.
@@ -178,7 +178,7 @@ This is polynomial, therefore holomorphic.
 So
 
 $$
-\int_{R} e^{\zeta^2}d\zeta = 0
+\int_{R} e^{z^2}dz = 0
 $$
 
 with some limit calculation, we can get
@@ -186,5 +186,5 @@ with some limit calculation, we can get
 <!--TODO: Fill the parts-->
 
 $$
-\int_{R} e^{-\zeta^2}d\zeta = 2\pi i
+\int_{R} e^{-z^2}dz = 2\pi i
 $$