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pages/Math416/Math416_L6.md

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+# Lecture 6
+
+## Review
+
+### Linear Fractional Transformations
+
+Transformations of the form $f(z)=\frac{az+b}{cz+d}$,$a,b,c,d\in\mathbb{C}$ and $ad-bc\neq 0$ are called linear fractional transformations.
+
+#### Theorem 3.8 Preservation of clircles
+
+We defined clircle to be a circle or a line.
+
+The circle equation is:
+
+Let $\zeta=u+iv$ be the center of the circle, $r$ be the radius of the circle.
+
+$$
+circle=\{z\in\mathbb{C}:|\zeta-c|=r\}
+$$
+
+This is:
+
+$$
+|\zeta|^2-c\overline{\zeta}-\overline{c}\zeta+|c|^2-r^2=0
+$$
+
+If $\phi$ is a non-constant linear fractional transformation, then $\phi$ maps clircles to clircles.
+
+We claim that a map is circle preserving if and only if for some $\alpha,\beta,\gamma,\delta\in\mathbb{R}$.
+
+$$
+\alpha|\zeta|^2+\beta Re(\zeta)+\gamma Im(\zeta)+\delta=0
+$$
+
+when $\alpha=0$, it is a line.
+
+when $\alpha\neq 0$, it is a circle.
+
+Proof:
+
+Let $w=u+iv=\frac{1}{\zeta}$, so $\frac{1}{w}=\frac{u}{u^2+v^2}-i\frac{v}{u^2+v^2}$.
+
+Then the original equation becomes:
+
+$$
+\alpha\left(\frac{u}{u^2+v^2}\right)^2+\beta\left(\frac{u}{u^2+v^2}\right)+\gamma\left(-\frac{v}{u^2+v^2}\right)+\delta=0
+$$
+
+Which is in the form of circle equation.
+
+EOP
+
+## Chapter 4 Elements of functions
+
+> $e^t=\sum_{n=0}^{\infty}\frac{t^n}{n!}$
+
+So, following the definition of $e^\zeta$, we have:
+
+$$
+\begin{aligned}
+e^{x+iy}&=e^xe^{iy} \\
+&=e^x\left(\sum_{n=0}^{\infty}\frac{(iy)^n}{n!}\right) \\
+&=e^x\left(\sum_{n=0}^{\infty}\frac{(-1)^ny^n}{n!}\right) \\
+&=e^x(\cos y+i\sin y)
+\end{aligned}
+$$
+
+### $e^\zeta$
+
+The exponential of $e^\zeta=x+iy$ is defined as:
+
+$$
+e^\zeta=exp(\zeta)=e^x(\cos y+i\sin y)
+$$
+
+So,
+
+$$
+|e^\zeta|=|e^x||\cos y+i\sin y|=e^x
+$$
+
+#### Theorem 4.3 $e^\zeta$ is holomorphic
+
+$e^\zeta$ is holomorphic on $\mathbb{C}$.
+
+Proof:
+
+$$
+\begin{aligned}
+\frac{\partial}{\partial\zeta}e^\zeta&=\frac{1}{2}\left(\frac{\partial}{\partial x}+\frac{i}{\partial y}\right)e^x(\cos y+i\sin y) \\
+&=\frac{1}{2}e^x(\cos y+i\sin y)+ie^x(-\sin y+i\cos y) \\
+&=0
+\end{aligned}
+$$
+
+EOP
+
+#### Theorem 4.4 $e^\zeta$ is periodic
+
+$e^\zeta$ is periodic with period $2\pi i$.
+
+Proof:
+
+$$
+e^{\zeta+2\pi i}=e^\zeta e^{2\pi i}=e^\zeta\cdot 1=e^\zeta
+$$
+
+EOP
+
+#### Theorem 4.5 $e^\zeta$ as a map
+
+$e^\zeta$ is a map from $\mathbb{C}$ to $\mathbb{C}$ with period $2\pi i$.
+
+$$
+e^{\pi i}+1=0
+$$
+
+This is a map from cartesian coordinates to polar coordinates, where $e^x$ is the radius and $y$ is the angle.
+
+This map attains every value in $\mathbb{C}\setminus\{0\}$.
+
+#### Definition 4.6-8 $\cos\zeta$ and $\sin\zeta$
+
+$$
+\cos\zeta=\frac{1}{2}(e^{i\zeta}+e^{-i\zeta})
+$$
+
+$$
+\sin\zeta=\frac{1}{2i}(e^{i\zeta}-e^{-i\zeta})
+$$
+
+$$
+\cosh\zeta=\frac{1}{2}(e^\zeta+e^{-\zeta})
+$$
+
+$$
+\sinh\zeta=\frac{1}{2}(e^\zeta-e^{-\zeta})
+$$
+
+From this definition, we can see that $\cos\zeta$ and $\sin\zeta$ are no longer bounded.
+
+And this definition is still compatible with the previous definition of $\cos$ and $\sin$ when $\zeta$ is real.
+
+Moreover,
+
+$$
+\cosh(i\zeta)=\cos\zeta
+$$
+
+$$
+\sinh(i\zeta)=i\sin\zeta
+$$
+
+### Logarithm
+
+#### Definition 4.9 Logarithm
+
+A logarithm of $a$ is any $b$ such that $e^b=a$.
+
+If $a=0$, then no logarithm exists.
+
+If $a\neq 0$, then there exists infinitely many logarithms of $a$.
+
+Let $a=re^{i\theta}$, $b=x+iy$ be a logarithm of $a$.
+
+Then,
+
+$$
+e^{x+iy}=re^{i\theta}
+$$
+
+Since logarithm is not unique, we can always add $2k\pi i$ to the angle.
+
+If $y\in(-\pi,\pi]$, then $\log a=b$ means $e^b=a$ and $Im(b)\in(-\pi,\pi]$.
+
+If $a=re^{i\theta}$, then $\log a=\log r+i(\theta_0+2k\pi)$.
+
+#### Definition 4.10
+
+Let $G$ be an open connected subset of $\mathbb{C}\setminus\{0\}$.
+
+A branch of $\arg(\zeta)$ in $G$ is a continuous function $\alpha$, such that $\alpha(\zeta)$ is a value of $\arg(\zeta)$.
+
+A branch of $\log(\zeta)$ in $G$ is a continuous function $\beta$, such that $e^{\beta(\zeta)}=\zeta$.
+
+Note: $G$ has a branch of $\arg(\zeta)$ if and only if it has a branch of $\log(\zeta)$.
+
+If $G=\mathbb{C}\setminus\{0\}$, then  not branch of $\arg(\zeta)$ exists.
+
+Suppose $\alpha_1$ and $\alpha_2$ are two branches of $\arg(\zeta)$ in $G$.
+
+Then,
+
+$$
+\alpha_1(\zeta)-\alpha_2(\zeta)=2k\pi i
+$$
+
+for some $k\in\mathbb{Z}$.
+
+#### Theorem 4.11
+
+$\log(\zeta)$ is holomorphic on $\mathbb{C}\setminus\{0\}$.
+
+Proof:
+
+Method 1: Use polar coordinates. (See in homework)
+
+Method 2: Use the fact that $\log(\zeta)$ is the inverse of $e^\zeta$.
+
+Suppose $h=s+it$, $e^h=e^s(\cos t+i\sin t)$, $e^h-1=e^s(\cos t-1)+i\sin t$. So
+
+$$
+\begin{aligned}
+\frac{e^h-1}{h}&=\frac{(s+it)e^s(\cos t-1)+i\sin t}{s^2+t^2} \\
+&=\frac{e^s(\cos t-1)}{s^2+t^2}+i\frac{\sin t}{s^2+t^2}
+\end{aligned}
+$$
+
+Continue next time.

pages/Math416/_meta.js

@@ -8,4 +8,5 @@ export default {
     Math416_L3: "Complex Variables (Lecture 3)",
     Math416_L4: "Complex Variables (Lecture 4)",
     Math416_L5: "Complex Variables (Lecture 5)",
+    Math416_L6: "Complex Variables (Lecture 6)",
 }