update and fix typo

pages/Math416/Math416_L10.md

@@ -40,7 +40,7 @@ EOP
 
 ### Complex Integral
 
-#### Definition 4.1
+#### Definition 6.1
 
 If $\phi(t)$ is a complex function defined on $[a,b]$, then the integral of $\phi(t)$ over $[a,b]$ is defined as
 
@@ -48,7 +48,7 @@ $$
 \int_a^b \phi(t) dt = \int_a^b \text{Re}\{\phi(t)\} dt + i\int_a^b \text{Im}\{\phi(t)\} dt
 $$
 
-#### Theorem 4.3 (Triangle Inequality)
+#### Theorem 6.3 (Triangle Inequality)
 
 If $\phi(t)$ is a complex function defined on $[a,b]$, then
 
@@ -74,7 +74,7 @@ Assume $\phi$ is continuous on $[a,b]$, the equality means $\lambda(t)\phi(t)$ i
 
 EOP
 
-#### Definition 4.4 Arc Length
+#### Definition 6.4 Arc Length
 
 Let $\gamma$ be a curve in the complex plane defined by $\gamma(t)=x(t)+iy(t)$, $t\in[a,b]$. The arc length of $\gamma$ is given by
 
@@ -117,7 +117,7 @@ $$
 \int_{\Gamma} f(\zeta) d\zeta=\int_0^{2\pi} f(\gamma(t))\gamma'(t) dt=\int_0^{2\pi} \frac{1}{Re^{-it}}iRe^{it} dt=2\pi i
 $$
 
-#### Theorem 4.11 (Uniform Convergence)
+#### Theorem 6.11 (Uniform Convergence)
 
 If $f_n(z)$ converges uniformly to $f(z)$ on $\Gamma$, assume length of $\Gamma$ is finite, then
 
@@ -144,7 +144,7 @@ $$
 
 EOP
 
-#### Theorem 4.6 (Integral of derivative)
+#### Theorem 6.6 (Integral of derivative)
 
 Suppose $\Gamma$ is a closed curve, $\gamma:[a,b]\to\mathbb{C}$ and $\gamma(a)=\gamma(b)$.
 

pages/Math416/Math416_L11.md

@@ -0,0 +1,121 @@
+# Math416 Lecture 11
+
+## Continue on integration over complex plane
+
+### Continue on last example
+
+Last lecture we have:Let $R$ be a rectangular start from the $-a$ to $a$, $a+ib$ to $-a+ib$, $\int_{R} e^{-\zeta^2}d\zeta=0$, however, the integral consists of four parts:
+
+Path 1: $-a\to a$
+
+$\int_{I_1}e^{-\zeta^2}d\zeta=\int_{-a}^{a}e^{-\zeta^2}d\zeta=\int_{-a}^{a}e^{-x^2}dx$
+
+Path 2: $a+ib\to -a+ib$
+
+$\int_{I_2}e^{-\zeta^2}d\zeta=\int_{a+ib}^{-a+ib}e^{-\zeta^2}d\zeta=\int_{0}^{b}e^{-(a+iy)^2}dy$
+
+Path 3: $-a+ib\to -a-ib$
+
+$-\int_{I_3}e^{-\zeta^2}d\zeta=-\int_{-a+ib}^{-a-ib}e^{-\zeta^2}d\zeta=-\int_{a}^{-a}e^{-(x-ib)^2}dx$
+
+Path 4: $-a-ib\to a-ib$
+
+$-\int_{I_4}e^{-\zeta^2}d\zeta=-\int_{-a-ib}^{a-ib}e^{-\zeta^2}d\zeta=-\int_{b}^{0}e^{-(-a+iy)^2}dy$
+
+> #### The reverse of a curve 6.9
+>
+> If $\gamma:[a,b]\to\mathbb{C}$ is a curve, then the rever of $\gamma$ is the curve $-\gamma:[-b,-a]\to\mathbb{C}$ defined by $(-\gamma)(t)=\gamma(a+b-t)$. It is the curve one obtains from $\gamma$ by traversing it in the opposite direction.
+>
+> - If $\gamma$ is piecewise in $C^1$, then $-\gamma$ is piecewise in $C^1$.
+> - $\int_{-\gamma}f(z)dz=-\int_{\gamma}f(z)dz$ for any function $f$ that is continuous on $\gamma([a,b])$.
+
+If we keep $b$ fixed, and let $a\to\infty$, then
+
+> #### Definition 6.10 (Estimate of the integral)
+>
+> Let $\gamma:[a,b]\to\mathbb{C}$ be a piecewise $C^1$ curve, and let $f:[a,b]\to\mathbb{C}$ be a continuous complex-valued function. Let $M$ be the maximum of $|f|$ on $\gamma([a,b])$. ($M=\max\{|f(t)|:t\in[a,b]\}$)
+>
+> Then
+>
+> $$\left|\int_{\gamma}f(\zeta)d\zeta\right|\leq L(\gamma)M$$
+
+_Continue on previous example, we have:_
+
+$$
+\left|\int_{\gamma}f(\zeta)d\zeta\right|\leq L(\gamma)\max_{\zeta\in\gamma}|f(\zeta)|\to 0
+$$
+
+Since,
+
+$$
+\int_{-\infty}^{\infty}e^{-x^2}dx-\int_{-\infty}^{\infty}e^{-x^2+b^2}(\cos 2bx+i\sin 2bx)dx=0
+$$
+
+Since $\sin 2bx$ is odd, and $\cos 2bx$ is even, we have
+
+$$
+\int_{-\infty}^{\infty}e^{-x^2}dx=\int_{-\infty}^{\infty}e^{-x^2+b^2}\cos 2bxdx=\sqrt{\pi}e^{-b^2}
+$$
+
+##### Proof for the last step:
+
+$$
+\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}
+$$
+
+Proof:
+
+Let $J=\int_{-\infty}^{\infty}e^{-x^2}dx$
+
+Then
+
+$$J^2=\int_{-\infty}^{\infty}e^{-x^2}dx\int_{-\infty}^{\infty}e^{-y^2}dy=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dxdy$$
+
+We can evaluate the integral on the right-hand side by converting to polar coordinates. $x=r\cos\theta$, $y=r\sin\theta,dxdy=rdrd\theta$
+
+$$
+J^2=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dxdy=\int_{0}^{2\pi}\int_{0}^{\infty}e^{-r^2}rdrd\theta
+$$
+
+$$
+J^2=\int_{0}^{2\pi}\int_{0}^{\infty}e^{-r^2}rdrd\theta=\int_{0}^{2\pi}\left[-\frac{1}{2}e^{-r^2}\right]_{0}^{\infty}d\theta
+$$
+
+$$
+J^2=\int_{0}^{2\pi}\frac{1}{2}d\theta=\pi
+$$
+
+$$
+J=\sqrt{\pi}
+$$
+
+EOP
+
+## Chapter 7 Cauchy's theorem
+
+### Cauchy's theorem (Fundamental theorem of complex function theory)
+
+Let $\gamma$ be a closed curve in $\mathbb{C}$ and let $u$ be an open set _containing $\gamma^*$_. Let $f$ be a holomorphic function on $u$. Then
+
+$$
+\int_{\gamma}f(z)dz=0
+$$
+
+Note: What "containing $\gamma^*$" means? (Rabbit hole for topologists)
+
+#### Lemma 7.1 (Goursat's lemma)
+
+Cauchy's theorem is true if $\gamma$ is a triangle.
+
+Proof:
+
+We plan to keep shrinking the triangle until $f(\zeta+h)=f(\zeta)+hf'(\zeta)+\mu(h)$ where $\mu(h)$ is a function of $h$ that goes to $0$ as $h\to 0$.
+
+Let's start with a triangle $T$ with vertices $z_1,z_2,z_3$.
+
+![carving a triangle](https://notenextra.trance-0.com/Math416/Cauchy_theorem_triangle_carving.png)
+
+We divide $T$ into four smaller triangles by drawing lines from the midpoints of the sides to the opposite vertices.
+
+
+

pages/Math416/_meta.js

@@ -13,4 +13,5 @@ export default {
     Math416_L8: "Complex Variables (Lecture 8)",
     Math416_L9: "Complex Variables (Lecture 9)",
     Math416_L10: "Complex Variables (Lecture 10)",
+    Math416_L11: "Complex Variables (Lecture 11)",
 }