Complex Variables & Transforms

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1. Complex Numbers Definition: $z = x + iy$, where $x, y \in \mathbb{R}$ and $i = \sqrt{-1}$. Real Part: $\text{Re}(z) = x$ Imaginary Part: $\text{Im}(z) = y$ Conjugate: $\bar{z} = x - iy$ Modulus (Magnitude): $|z| = \sqrt{x^2 + y^2} = \sqrt{z\bar{z}}$ Argument (Angle): $\arg(z) = \theta$ where $x = |z|\cos\theta$, $y = |z|\sin\theta$. Principal value: $-\pi Polar Form: $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$, where $r = |z|$. Euler's Formula: $e^{i\theta} = \cos\theta + i\sin\theta$ De Moivre's Theorem: $(re^{i\theta})^n = r^n e^{in\theta} = r^n(\cos(n\theta) + i\sin(n\theta))$ 2. Complex Functions Definition: $w = f(z) = u(x,y) + iv(x,y)$, where $u,v$ are real functions. Analytic Function: A function $f(z)$ is analytic at a point $z_0$ if it is differentiable not only at $z_0$ but at every point in some neighborhood of $z_0$. Cauchy-Riemann Equations: For $f(z) = u(x,y) + iv(x,y)$ to be analytic, its partial derivatives must satisfy: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$ In polar coordinates: $\frac{\partial u}{\partial r} = \frac{1}{r}\frac{\partial v}{\partial \theta}$ $\frac{1}{r}\frac{\partial u}{\partial \theta} = -\frac{\partial v}{\partial r}$ Harmonic Function: A real function $\phi(x,y)$ is harmonic if it satisfies Laplace's equation: $\nabla^2\phi = \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0$. Harmonic Conjugate: If $f(z) = u+iv$ is analytic, then $u$ and $v$ are harmonic conjugates of each other. 3. Elementary Complex Functions Exponential Function: $e^z = e^{x+iy} = e^x(\cos y + i\sin y)$ Logarithm: $\ln z = \ln|z| + i\arg(z) = \ln r + i(\theta + 2k\pi)$, $k \in \mathbb{Z}$. Principal Value: $\text{Ln } z = \ln r + i\theta$, where $-\pi Trigonometric Functions: $\cos z = \frac{e^{iz} + e^{-iz}}{2}$ $\sin z = \frac{e^{iz} - e^{-iz}}{2i}$ Hyperbolic Functions: $\cosh z = \frac{e^z + e^{-z}}{2}$ $\sinh z = \frac{e^z - e^{-z}}{2}$ Relation: $\cosh(iz) = \cos z$, $\sinh(iz) = i\sin z$ Complex Powers: $z^c = e^{c \ln z}$ 4. Complex Integration Line Integral: $\int_C f(z) dz = \int_a^b f(z(t))z'(t) dt$ Cauchy's Integral Theorem: If $f(z)$ is analytic within and on a simple closed contour $C$, then $\oint_C f(z) dz = 0$. Cauchy's Integral Formula: If $f(z)$ is analytic within and on a simple closed contour $C$ and $z_0$ is any point inside $C$, then $f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-z_0} dz$. Cauchy's Integral Formula for Derivatives: $f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z-z_0)^{n+1}} dz$. 5. Series Taylor Series: If $f(z)$ is analytic at $z_0$, then $f(z) = \sum_{n=0}^{\infty} a_n (z-z_0)^n$, where $a_n = \frac{f^{(n)}(z_0)}{n!}$. Maclaurin Series: Taylor series with $z_0=0$. Laurent Series: If $f(z)$ is analytic in an annulus $R_1 6. Singularities and Residues Isolated Singularity: A point $z_0$ where $f(z)$ is not analytic but is analytic in a punctured disk $0 Types of Isolated Singularities: Removable: Laurent series has no negative powers. $\lim_{z \to z_0} f(z)$ exists. Pole of Order $m$: Laurent series has a finite number of negative powers, with $a_{-m} \neq 0$. $\lim_{z \to z_0} (z-z_0)^m f(z)$ exists and is non-zero. Essential: Laurent series has infinitely many negative powers. $\lim_{z \to z_0} f(z)$ does not exist. Residue: For an isolated singularity $z_0$, the residue is the coefficient $a_{-1}$ of the Laurent series. $\text{Res}(f, z_0) = a_{-1}$. Formulas for Residues: Simple Pole ($m=1$): $\text{Res}(f, z_0) = \lim_{z \to z_0} (z-z_0)f(z)$. Pole of Order $m$: $\text{Res}(f, z_0) = \frac{1}{(m-1)!} \lim_{z \to z_0} \frac{d^{m-1}}{dz^{m-1}} [(z-z_0)^m f(z)]$. Cauchy's Residue Theorem: If $f(z)$ is analytic inside and on a simple closed contour $C$, except for a finite number of isolated singularities $z_k$ inside $C$, then $\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$. 7. Laplace Transform Definition: $\mathcal{L}\{f(t)\} = F(s) = \int_0^{\infty} e^{-st} f(t) dt$ Inverse Laplace Transform: $\mathcal{L}^{-1}\{F(s)\} = f(t) = \frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} e^{st} F(s) ds$ (Bromwich Integral) Properties of Laplace Transform Property Time Domain $f(t)$ Frequency Domain $F(s)$ Linearity $af(t) + bg(t)$ $aF(s) + bG(s)$ Time Scaling $f(at)$ $\frac{1}{a}F(\frac{s}{a})$ Frequency Shifting $e^{at}f(t)$ $F(s-a)$ Time Shifting $f(t-a)u(t-a)$ $e^{-as}F(s)$ Differentiation (time) $f'(t)$ $sF(s) - f(0)$ Integration (time) $\int_0^t f(\tau) d\tau$ $\frac{1}{s}F(s)$ Multiplication by $t^n$ $t^n f(t)$ $(-1)^n \frac{d^n}{ds^n} F(s)$ Division by $t$ $\frac{f(t)}{t}$ $\int_s^{\infty} F(\sigma) d\sigma$ Convolution $(f*g)(t) = \int_0^t f(\tau)g(t-\tau)d\tau$ $F(s)G(s)$ Common Laplace Transform Pairs $f(t)$ $F(s) = \mathcal{L}\{f(t)\}$ $\delta(t)$ (Dirac Delta) $1$ $u(t)$ (Unit Step) $\frac{1}{s}$ $t^n$, $n \in \mathbb{Z}_{\ge 0}$ $\frac{n!}{s^{n+1}}$ $e^{at}$ $\frac{1}{s-a}$ $\sin(at)$ $\frac{a}{s^2+a^2}$ $\cos(at)$ $\frac{s}{s^2+a^2}$ $\sinh(at)$ $\frac{a}{s^2-a^2}$ $\cosh(at)$ $\frac{s}{s^2-a^2}$ $t e^{at}$ $\frac{1}{(s-a)^2}$ 8. Fourier Series Periodic Function: $f(t)$ with period $T$, $f(t+T) = f(t)$. Angular frequency $\omega_0 = \frac{2\pi}{T}$. Trigonometric Fourier Series: $f(t) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t))$ $a_0 = \frac{1}{T} \int_0^T f(t) dt$ $a_n = \frac{2}{T} \int_0^T f(t) \cos(n\omega_0 t) dt$ $b_n = \frac{2}{T} \int_0^T f(t) \sin(n\omega_0 t) dt$ Complex Exponential Fourier Series: $f(t) = \sum_{n=-\infty}^{\infty} c_n e^{in\omega_0 t}$ $c_n = \frac{1}{T} \int_0^T f(t) e^{-in\omega_0 t} dt$ Relationship: $c_0 = a_0$, $c_n = \frac{1}{2}(a_n - ib_n)$, $c_{-n} = \frac{1}{2}(a_n + ib_n)$ Parseval's Theorem: $\frac{1}{T} \int_0^T |f(t)|^2 dt = \sum_{n=-\infty}^{\infty} |c_n|^2 = a_0^2 + \frac{1}{2}\sum_{n=1}^{\infty} (a_n^2 + b_n^2)$ 9. Fourier Transform Definition: $\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$ Inverse Fourier Transform: $\mathcal{F}^{-1}\{F(\omega)\} = f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{i\omega t} d\omega$ Properties of Fourier Transform Property Time Domain $f(t)$ Frequency Domain $F(\omega)$ Linearity $af(t) + bg(t)$ $aF(\omega) + bG(\omega)$ Time Shifting $f(t-t_0)$ $e^{-i\omega t_0}F(\omega)$ Frequency Shifting $e^{i\omega_0 t}f(t)$ $F(\omega-\omega_0)$ Time Scaling $f(at)$ $\frac{1}{|a|}F(\frac{\omega}{a})$ Duality $F(t)$ $2\pi f(-\omega)$ Differentiation (time) $\frac{d}{dt}f(t)$ $i\omega F(\omega)$ Integration (time) $\int_{-\infty}^t f(\tau) d\tau$ $\frac{1}{i\omega}F(\omega) + \pi F(0)\delta(\omega)$ Convolution $(f*g)(t)$ $F(\omega)G(\omega)$ Multiplication $f(t)g(t)$ $\frac{1}{2\pi}(F*G)(\omega)$ Common Fourier Transform Pairs $f(t)$ $F(\omega) = \mathcal{F}\{f(t)\}$ $\delta(t)$ $1$ $1$ $2\pi\delta(\omega)$ $e^{iat}$ $2\pi\delta(\omega-a)$ $\cos(a t)$ $\pi[\delta(\omega-a) + \delta(\omega+a)]$ $\sin(a t)$ $-i\pi[\delta(\omega-a) - \delta(\omega+a)]$ $e^{-a|t|}$, $a>0$ $\frac{2a}{\omega^2+a^2}$ $e^{-at}u(t)$, $a>0$ $\frac{1}{a+i\omega}$ Rect$(t/T)$ $T \text{sinc}(\frac{\omega T}{2})$ Tri$(t/T)$ $T \text{sinc}^2(\frac{\omega T}{2})$