# Math 416 Midterm 1 Review So everything we have learned so far is to extend the real line to the complex plane. ## Chapter 1 Complex Numbers ### Definition of complex numbers An ordered pair of real numbers $(x, y)$ can be represented as a complex number $z = x + yi$, where $i$ is the imaginary unit. With operations defined as: $$ (x_1 + y_1i) + (x_2 + y_2i) = (x_1 + x_2) + (y_1 + y_2)i $$ $$ (x_1 + y_1i) \cdot (x_2 + y_2i) = (x_1x_2 - y_1y_2) + (x_1y_2 + x_2y_1)i $$ ### De Moivre's Formula Every complex number $z$ can be written as $z = r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude of $z$ and $\theta$ is the argument of $z$. $$ z^n = r^n(\cos n\theta + i \sin n\theta) $$ The De Moivre's formula is useful for finding the $n$th roots of a complex number. $$ z^n = r^n(\cos n\theta + i \sin n\theta) $$ ### Roots of complex numbers Using De Moivre's formula, we can find the $n$th roots of a complex number. If $z=r(\cos \theta + i \sin \theta)$, then the $n$th roots of $z$ are given by: $$ z_k = r^{1/n}(\cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n}) $$ for $k = 0, 1, 2, \ldots, n-1$. ### Stereographic projection ![Stereographic projection](https://notenextra.trance-0.com/Math416/Stereographic_projection.png) The stereographic projection is a map from the unit sphere $S^2$ to the complex plane $\mathbb{C}\setminus\{0\}$. The projection is given by: $$ z\mapsto \frac{(2Re(z), 2Im(z), |z|^2-1)}{|z|^2+1} $$ The inverse map is given by: $$ (\xi,\eta, \zeta)\mapsto \frac{\xi + i\eta}{1 - \zeta} $$ ## Chapter 2 Complex Differentiation ### Definition of complex differentiation Let the complex plane $\mathbb{C}$ be defined in an open subset $G$ of $\mathbb{C}$. (Domain) Then $f$ is said to be differentiable at $z_0\in G$ if the limit $$ \lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0} $$ exists. The limit is called the derivative of $f$ at $z_0$ and is denoted by $f'(z_0)$. To prove that a function is differentiable, we can use the standard delta-epsilon definition of a limit. $$ \left|\frac{f(z)-f(z_0)}{z-z_0} - f'(z_0)\right| < \epsilon $$ whenever $0 < |z-z_0| < \delta$. With such definition, all the properties of real differentiation can be extended to complex differentiation. #### Differentiation of complex functions 1. If $f$ is differentiable at $z_0$, then $f$ is continuous at $z_0$. 2. If $f,g$ are differentiable at $z_0$, then $f+g, fg$ are differentiable at $z_0$. $$ (f+g)'(z_0) = f'(z_0) + g'(z_0) $$ $$ (fg)'(z_0) = f'(z_0)g(z_0) + f(z_0)g'(z_0) $$ 3. If $f,g$ are differentiable at $z_0$ and $g(z_0)\neq 0$, then $f/g$ is differentiable at $z_0$. $$ \left(\frac{f}{g}\right)'(z_0) = \frac{f'(z_0)g(z_0) - f(z_0)g'(z_0)}{g(z_0)^2} $$ 4. If $f$ is differentiable at $z_0$ and $g$ is differentiable at $f(z_0)$, then $g\circ f$ is differentiable at $z_0$. $$ (g\circ f)'(z_0) = g'(f(z_0))f'(z_0) $$ 5. If $f(z)=\sum_{k=0}^n c_k(z-z_0)^k$, where $c_k\in\mathbb{C}$, then $f$ is differentiable at $z_0$ and $f'(z_0)=\sum_{k=1}^n kc_k(z_0-z_0)^{k-1}$. $$ f'(z_0) = c_1 + 2c_2(z_0-z_0) + 3c_3(z_0-z_0)^2 + \cdots + nc_n(z_0-z_0)^{n-1} $$ ### Cauchy-Riemann Equations Let the function defined on an open subset $G$ of $\mathbb{C}$ be $f(x,y)=u(x,y)+iv(x,y)$, where $u,v$ are real-valued functions. Then $f$ is differentiable at $z_0=x_0+y_0i$ if and only if the partial derivatives of $u$ and $v$ exist at $(x_0,y_0)$ and satisfy the Cauchy-Riemann equations: $$ \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} $$ ### Holomorphic functions A function $f$ is said to be holomorphic on an open subset $G$ of $\mathbb{C}$ if $f$ is differentiable at every point of $G$. #### Partial differential operators $$ \frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}\right) $$ $$ \frac{\partial}{\partial \bar{z}} = \frac{1}{2}\left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y}\right) $$ This gives that $$ \frac{\partial f}{\partial z} = \frac{1}{2}\left(\frac{\partial f}{\partial x} - i\frac{\partial f}{\partial y}\right)=\frac{1}{2}\left(\frac{\partial u}{\partial x} +\frac{\partial v}{\partial y}\right) + \frac{i}{2}\left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right) $$ $$ \frac{\partial f}{\partial \bar{z}} = \frac{1}{2}\left(\frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}\right)=\frac{1}{2}\left(\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}\right) + \frac{i}{2}\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right) $$ If the function $f$ is holomorphic, then by the Cauchy-Riemann equations, we have $$ \frac{\partial f}{\partial \bar{z}} = 0 $$ ### Conformal mappings A holomorphic function $f$ is said to be conformal if it preserves the angles between the curves. More formally, if $f$ is holomorphic on an open subset $G$ of $\mathbb{C}$ and $z_0\in G$, $\gamma_1, \gamma_2$ are two curves passing through $z_0$ ($\gamma_1(t_1)=\gamma_2(t_2)=z_0$) and intersecting at an angle $\theta$, then $$ \arg(f\circ\gamma_1)'(t_1) - \arg(f\circ\gamma_2)'(t_2) = \theta $$ In other words, the angle between the curves is preserved. An immediate consequence is that $$ \arg(f\cdot \gamma_1)'(t_1) =\arg f'(z_0) + \arg \gamma_1'(t_1)\\ \arg(f\cdot \gamma_2)'(t_2) =\arg f'(z_0) + \arg \gamma_2'(t_2) $$ ### Harmonic functions A real-valued function $u$ is said to be harmonic if it satisfies the Laplace equation: $$ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 $$ ## Chapter 3 Linear Fractional Transformations ### Definition of linear fractional transformations A linear fractional transformation is a function of the form $$ \phi(z) = \frac{az+b}{cz+d} $$ where $a,b,c,d$ are complex numbers and $ad-bc\neq 0$. ### Properties of linear fractional transformations #### Conformality A linear fractional transformation is conformal. $$ \phi'(z) = \frac{ad-bc}{(cz+d)^2} $$ #### Three-fold transitivity If $z_1,z_2,z_3$ are distinct points in the complex plane, then there exists a unique linear fractional transformation $\phi$ such that $\phi(z_1)=\infty$, $\phi(z_2)=0$, $\phi(z_3)=1$. The map is given by $$ \phi(z) =\begin{cases} \frac{(z-z_2)(z_1-z_3)}{(z-z_1)(z_2-z_3)} & \text{if } z_1,z_2,z_3 \text{ are all finite}\\ \frac{z-z_2}{z_3-z_2} & \text{if } z_1=\infty\\ \frac{z_3-z_1}{z-z_1} & \text{if } z_2=\infty\\ \frac{z-z_2}{z-z_1} & \text{if } z_3=\infty\\ \end{cases} $$ So if $z_1,z_2,z_3$, $w_1,w_2,w_3$ are distinct points in the complex plane, then there exists a unique linear fractional transformation $\phi$ such that $\phi(z_i)=w_i$ for $i=1,2,3$. #### Inversion #### Factorization #### Clircle ## Chapter 4 Elementary Functions ### Exponential function ### Trigonometric functions ### Logarithmic function ### Power function ### Inverse trigonometric functions ## Chapter 5 Power Series ### Definition of power series ### Properties of power series ### Radius/Region of convergence ### Cauchy-Hadamard Theorem ### Cauchy Product (of power series) ## Chapter 6 Complex Integration ### Definition of Riemann Integral for complex functions The complex integral of a complex function $\phi$ on the closed subinterval $[a,b]$ of the real line is said to be piecewise continuous if there exists a partition $a=t_0