304 lines
8.3 KiB
Markdown
304 lines
8.3 KiB
Markdown
# Math 416 Midterm 1 Review
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So everything we have learned so far is to extend the real line to the complex plane.
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## Chapter 1 Complex Numbers
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### Definition of complex numbers
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An ordered pair of real numbers $(x, y)$ can be represented as a complex number $z = x + yi$, where $i$ is the imaginary unit.
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With operations defined as:
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$$
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(x_1 + y_1i) + (x_2 + y_2i) = (x_1 + x_2) + (y_1 + y_2)i
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$$
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$$
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(x_1 + y_1i) \cdot (x_2 + y_2i) = (x_1x_2 - y_1y_2) + (x_1y_2 + x_2y_1)i
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$$
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### De Moivre's Formula
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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 $z$.
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$$
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z^n = r^n(\cos n\theta + i \sin n\theta)
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$$
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The De Moivre's formula is useful for finding the $n$th roots of a complex number.
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$$
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z^n = r^n(\cos n\theta + i \sin n\theta)
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$$
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### Roots of complex numbers
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Using De Moivre's formula, we can find the $n$th roots of a complex number.
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If $z=r(\cos \theta + i \sin \theta)$, then the $n$th roots of $z$ are given by:
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$$
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z_k = r^{1/n}(\cos \frac{\theta + 2k\pi}{n} + i \sin \frac{\theta + 2k\pi}{n})
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$$
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for $k = 0, 1, 2, \ldots, n-1$.
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### Stereographic projection
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The stereographic projection is a map from the unit sphere $S^2$ to the complex plane $\mathbb{C}\setminus\{0\}$.
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The projection is given by:
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$$
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z\mapsto \frac{(2Re(z), 2Im(z), |z|^2-1)}{|z|^2+1}
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$$
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The inverse map is given by:
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$$
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(\xi,\eta, \zeta)\mapsto \frac{\xi + i\eta}{1 - \zeta}
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$$
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## Chapter 2 Complex Differentiation
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### Definition of complex differentiation
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Let the complex plane $\mathbb{C}$ be defined in an open subset $G$ of $\mathbb{C}$. (Domain)
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Then $f$ is said to be differentiable at $z_0\in G$ if the limit
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$$
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\lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0}
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$$
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exists.
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The limit is called the derivative of $f$ at $z_0$ and is denoted by $f'(z_0)$.
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To prove that a function is differentiable, we can use the standard delta-epsilon definition of a limit.
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$$
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\left|\frac{f(z)-f(z_0)}{z-z_0} - f'(z_0)\right| < \epsilon
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$$
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whenever $0 < |z-z_0| < \delta$.
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With such definition, all the properties of real differentiation can be extended to complex differentiation.
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#### Differentiation of complex functions
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1. If $f$ is differentiable at $z_0$, then $f$ is continuous at $z_0$.
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2. If $f,g$ are differentiable at $z_0$, then $f+g, fg$ are differentiable at $z_0$.
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$$
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(f+g)'(z_0) = f'(z_0) + g'(z_0)
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$$
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$$
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(fg)'(z_0) = f'(z_0)g(z_0) + f(z_0)g'(z_0)
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$$
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3. If $f,g$ are differentiable at $z_0$ and $g(z_0)\neq 0$, then $f/g$ is differentiable at $z_0$.
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$$
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\left(\frac{f}{g}\right)'(z_0) = \frac{f'(z_0)g(z_0) - f(z_0)g'(z_0)}{g(z_0)^2}
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$$
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4. If $f$ is differentiable at $z_0$ and $g$ is differentiable at $f(z_0)$, then $g\circ f$ is differentiable at $z_0$.
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$$
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(g\circ f)'(z_0) = g'(f(z_0))f'(z_0)
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$$
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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 $f'(z_0)=\sum_{k=1}^n kc_k(z_0-z_0)^{k-1}$.
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$$
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f'(z_0) = c_1 + 2c_2(z_0-z_0) + 3c_3(z_0-z_0)^2 + \cdots + nc_n(z_0-z_0)^{n-1}
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$$
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### Cauchy-Riemann Equations
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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.
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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:
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$$
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\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
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$$
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### Holomorphic functions
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A function $f$ is said to be holomorphic on an open subset $G$ of $\mathbb{C}$ if $f$ is differentiable at every point of $G$.
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#### Partial differential operators
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$$
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\frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}\right)
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$$
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$$
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\frac{\partial}{\partial \bar{z}} = \frac{1}{2}\left(\frac{\partial}{\partial x} + i\frac{\partial}{\partial y}\right)
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$$
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This gives that
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$$
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\frac{\partial f}{\partial z} = \frac{1}{2}\left(\frac{\partial f}{\partial x} - i\frac{\partial f}{\partial y}\right)=\frac{1}{2}\left(\frac{\partial u}{\partial x} +\frac{\partial v}{\partial y}\right) + \frac{i}{2}\left(\frac{\partial v}{\partial x} - \frac{\partial u}{\partial y}\right)
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$$
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$$
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\frac{\partial f}{\partial \bar{z}} = \frac{1}{2}\left(\frac{\partial f}{\partial x} + i\frac{\partial f}{\partial y}\right)=\frac{1}{2}\left(\frac{\partial u}{\partial x} - \frac{\partial v}{\partial y}\right) + \frac{i}{2}\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x}\right)
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$$
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If the function $f$ is holomorphic, then by the Cauchy-Riemann equations, we have
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$$
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\frac{\partial f}{\partial \bar{z}} = 0
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$$
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### Conformal mappings
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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
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$$
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\arg(f\circ\gamma_1)'(t_1) - \arg(f\circ\gamma_2)'(t_2) = \theta
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$$
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In other words, the angle between the curves is preserved.
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An immediate consequence is that
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$$
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\arg(f\cdot \gamma_1)'(t_1) =\arg f'(z_0) + \arg \gamma_1'(t_1)\\
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\arg(f\cdot \gamma_2)'(t_2) =\arg f'(z_0) + \arg \gamma_2'(t_2)
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$$
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### Harmonic functions
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A real-valued function $u$ is said to be harmonic if it satisfies the Laplace equation:
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$$
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\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
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$$
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## Chapter 3 Linear Fractional Transformations
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### Definition of linear fractional transformations
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A linear fractional transformation is a function of the form
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$$
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\phi(z) = \frac{az+b}{cz+d}
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$$
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where $a,b,c,d$ are complex numbers and $ad-bc\neq 0$.
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### Properties of linear fractional transformations
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#### Conformality
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A linear fractional transformation is conformal.
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$$
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\phi'(z) = \frac{ad-bc}{(cz+d)^2}
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$$
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#### Three-fold transitivity
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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 $\phi(z_1)=\infty$, $\phi(z_2)=0$, $\phi(z_3)=1$.
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The map is given by
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$$
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\phi(z) =\begin{cases}
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\frac{(z-z_2)(z_1-z_3)}{(z-z_1)(z_2-z_3)} & \text{if } z_1,z_2,z_3 \text{ are all finite}\\
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\frac{z-z_2}{z_3-z_2} & \text{if } z_1=\infty\\
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\frac{z_3-z_1}{z-z_1} & \text{if } z_2=\infty\\
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\frac{z-z_2}{z-z_1} & \text{if } z_3=\infty\\
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\end{cases}
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$$
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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 $i=1,2,3$.
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#### Inversion
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#### Factorization
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#### Clircle
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## Chapter 4 Elementary Functions
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### Exponential function
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### Trigonometric functions
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### Logarithmic function
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### Power function
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### Inverse trigonometric functions
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## Chapter 5 Power Series
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### Definition of power series
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### Properties of power series
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### Radius/Region of convergence
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### Cauchy-Hadamard Theorem
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### Cauchy Product (of power series)
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## Chapter 6 Complex Integration
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### Definition of Riemann Integral for complex functions
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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 $a=t_0<t_1<\cdots<t_n=b$ such that $\phi$ is continuous on each open interval $(t_{i-1},t_i)$ and has a finite limit at each discontinuity point of the closed interval $[a,b]$.
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If $\phi$ is piecewise continuous on $[a,b]$, then the complex integral of $\phi$ on $[a,b]$ is defined as
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$$
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\int_a^b \phi(t) dt = \int_a^b \operatorname{Re}\phi(t) dt + i\int_a^b \operatorname{Im}\phi(t) dt
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$$
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### Fundamental Theorem of Calculus
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If $\phi$ is piecewise continuous on $[a,b]$, then
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$$
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\int_a^b \phi'(t) dt = \phi(b)-\phi(a)
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$$
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### Triangle inequality
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$$
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\left|\int_a^b \phi(t) dt\right| \leq \int_a^b |\phi(t)| dt
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$$
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### Integral on curve
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Let $\gamma$ be a piecewise smooth curve in the complex plane.
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The integral of a complex function $f$ on $\gamma$ is defined as
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$$
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\int_\gamma f(z) dz = \int_a^b f(\gamma(t))\gamma'(t) dt
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$$
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### Properties of complex integrals
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1. Linearity:
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## Chapter 7 Cauchy's Theorem
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### Cauchy's Theorem
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### Cauchy's Formula for a Circle
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### Cauchy's Product
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