1.5 Complex variables

Complex numbers extend real numbers by introducing an imaginary unit i, where i^2 = -1. They're crucial in control theory and signal processing, allowing representation of two-dimensional quantities like amplitude and phase in a single entity.

The complex plane visually represents complex numbers, with real and imaginary parts on horizontal and vertical axes. This graphical tool aids understanding of complex number properties and operations, essential for analyzing control systems.

Complex numbers

• Complex numbers extend the concept of real numbers by introducing an imaginary unit, denoted as $i$, where $i^2 = -1$
• Complex numbers are fundamental in various fields of mathematics, physics, and engineering, particularly in the study of control theory and signal processing
• Complex numbers allow for the representation and manipulation of two-dimensional quantities, such as amplitude and phase, in a single mathematical entity

Real and imaginary parts

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• A complex number $z$ is composed of a real part $a$ and an imaginary part $b$, written as $z = a + bi$
• The real part represents the horizontal component of the complex number on the complex plane
• The imaginary part represents the vertical component of the complex number on the complex plane
• The real and imaginary parts can be accessed using the functions $\Re(z)$ and $\Im(z)$, respectively

Modulus and argument

• The modulus (or absolute value) of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$, representing the distance from the origin to the point $(a, b)$ on the complex plane
• The argument (or phase) of a complex number $z = a + bi$ is given by $\arg(z) = \arctan(\frac{b}{a})$, representing the angle between the positive real axis and the line connecting the origin to the point $(a, b)$
• The modulus and argument provide a way to describe complex numbers in terms of their magnitude and direction

Polar and exponential forms

• The polar form of a complex number $z$ is given by $z = r(\cos\theta + i\sin\theta)$, where $r$ is the modulus and $\theta$ is the argument
• The exponential form of a complex number $z$ is given by $z = re^{i\theta}$, which is derived from Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$
• The polar and exponential forms simplify the multiplication, division, and exponentiation of complex numbers
• Converting between rectangular (a + bi), polar, and exponential forms is essential for various calculations in control theory

Complex plane

• The complex plane, also known as the Argand plane, is a two-dimensional representation of complex numbers
• The horizontal axis represents the real part of the complex number, while the vertical axis represents the imaginary part
• The complex plane provides a visual tool for understanding the properties and operations of complex numbers

Argand diagram

• An Argand diagram is a graphical representation of complex numbers on the complex plane
• Each complex number $z = a + bi$ is represented as a point $(a, b)$ on the plane
• The distance from the origin to the point $(a, b)$ represents the modulus of the complex number
• The angle between the positive real axis and the line connecting the origin to the point $(a, b)$ represents the argument of the complex number

Graphical representation of complex numbers

• Complex numbers can be represented graphically using vectors or points on the complex plane
• Vector representation: A complex number $z = a + bi$ is represented as a vector with its tail at the origin and its head at the point $(a, b)$
• Point representation: A complex number $z = a + bi$ is represented as a point $(a, b)$ on the complex plane
• Graphical representations help visualize complex number operations such as addition, subtraction, multiplication, and division

Complex functions

• Complex functions are functions that map complex numbers from one complex plane (the domain) to another complex plane (the codomain)
• Complex functions play a crucial role in control theory, as they are used to model and analyze various systems and signals
• Understanding the properties and behavior of complex functions is essential for designing and analyzing control systems

Analytic functions

• An analytic function (or holomorphic function) is a complex function that is differentiable at every point in its domain
• Analyticity is a stronger condition than differentiability for real functions, as it requires the function to be differentiable in a neighborhood of each point
• Examples of analytic functions include polynomials, exponential functions, and trigonometric functions
• Analytic functions have important properties, such as the ability to be represented by power series and the preservation of angles under conformal mapping

Cauchy-Riemann equations

• The Cauchy-Riemann equations are a set of partial differential equations that provide a necessary and sufficient condition for a complex function to be analytic
• For a complex function $f(z) = u(x, y) + iv(x, y)$, where $u$ and $v$ are real-valued functions, the Cauchy-Riemann equations are:
  • $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$
  • $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• If a complex function satisfies the Cauchy-Riemann equations, it is analytic (or holomorphic)
• The Cauchy-Riemann equations are used to determine the analyticity of complex functions and to find the derivative of an analytic function

Harmonic functions

• A harmonic function is a twice continuously differentiable real-valued function that satisfies Laplace's equation: $\nabla^2 f = 0$
• In the context of complex analysis, the real and imaginary parts of an analytic function are harmonic functions
• If $f(z) = u(x, y) + iv(x, y)$ is an analytic function, then both $u(x, y)$ and $v(x, y)$ are harmonic functions
• Harmonic functions have important properties, such as the mean value property and the maximum principle, which are useful in solving boundary value problems in control theory

Complex integration

• Complex integration extends the concept of integration to complex functions and complex domains
• Complex integration is a powerful tool for evaluating integrals of real functions by using the properties of analytic functions and the residue theorem
• In control theory, complex integration is used to analyze the stability and performance of systems, as well as to compute the frequency response of systems

Contour integrals

• A contour integral is an integral of a complex function along a curve (or contour) in the complex plane
• The contour integral of a complex function $f(z)$ along a curve $C$ is denoted as $\int_C f(z) dz$
• Contour integrals are used to evaluate integrals of complex functions and to derive important results in complex analysis, such as Cauchy's integral theorem and the residue theorem
• In control theory, contour integrals are used to compute the frequency response of systems and to analyze the stability of closed-loop systems

Cauchy's integral theorem

• Cauchy's integral theorem states that if $f(z)$ is an analytic function in a simply connected domain $D$, then the contour integral of $f(z)$ along any closed curve $C$ within $D$ is zero: $\oint_C f(z) dz = 0$
• Cauchy's integral theorem is a fundamental result in complex analysis, as it establishes the independence of the contour integral from the path chosen, provided the function is analytic in the domain
• The theorem is used to simplify the evaluation of contour integrals and to derive other important results, such as Cauchy's integral formula

Cauchy's integral formula

• Cauchy's integral formula is a consequence of Cauchy's integral theorem and provides a way to express the value of an analytic function at a point in terms of a contour integral
• For an analytic function $f(z)$ in a domain $D$ and a point $z_0$ within $D$, Cauchy's integral formula states that: $f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} dz$, where $C$ is any closed curve enclosing $z_0$
• Cauchy's integral formula is used to evaluate complex integrals, to derive the Taylor series expansion of analytic functions, and to solve boundary value problems in control theory

Residue theorem

• The residue theorem is a powerful tool in complex analysis that relates the contour integral of a meromorphic function (a function that is analytic except for poles) to the sum of its residues
• The residue theorem states that if $f(z)$ is a meromorphic function in a domain $D$ and $C$ is a closed curve within $D$ enclosing a set of poles $z_1, z_2, ..., z_n$, then: $\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$, where $\text{Res}(f, z_k)$ is the residue of $f(z)$ at the pole $z_k$
• The residue theorem simplifies the evaluation of complex integrals and is particularly useful for computing real integrals using contour integration techniques

Singularities and residues

• A singularity of a complex function $f(z)$ is a point where the function is not analytic (e.g., poles, essential singularities, and branch points)
• A pole of a complex function $f(z)$ is a singularity where the function becomes unbounded but remains analytic in a neighborhood of the point after multiplying by a suitable power of $(z - z_0)$
• The residue of a complex function $f(z)$ at a pole $z_0$ is the coefficient of the $(z - z_0)^{-1}$ term in the Laurent series expansion of $f(z)$ around $z_0$
• Residues are used in the residue theorem to evaluate complex integrals and to compute real integrals using contour integration

Calculation of residues

• The residue of a complex function $f(z)$ at a simple pole $z_0$ can be calculated using the formula: $\text{Res}(f, z_0) = \lim_{z \to z_0} (z - z_0) f(z)$
• For higher-order poles, the residue can be calculated using the formula: $\text{Res}(f, z_0) = \frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}} [(z - z_0)^n f(z)]$, where $n$ is the order of the pole
• In some cases, the residue can be calculated by expanding the function into a Laurent series and identifying the coefficient of the $(z - z_0)^{-1}$ term

Application to real integrals

• The residue theorem can be used to evaluate real integrals by extending the integrand to the complex plane and choosing an appropriate contour
• The contour is chosen such that the integral along the contour is equal to the desired real integral, and the contour integral is then evaluated using the residue theorem
• Common techniques for evaluating real integrals using the residue theorem include:
  • Integrating along the real axis and closing the contour in the upper or lower half-plane
  • Integrating along a semicircular contour and letting the radius tend to infinity
  • Integrating along a keyhole contour to handle branch cuts and singularities on the real axis

Conformal mapping

• Conformal mapping is a transformation that preserves angles between curves in the complex plane
• A conformal map is an analytic function that maps one complex domain to another while preserving the local geometry (angles and shapes) of the domain
• Conformal mapping is used in control theory to simplify the analysis and design of systems by transforming complex geometries into simpler ones

Preservation of angles

• The defining property of a conformal map is the preservation of angles between curves
• If two curves intersect at a point in the original domain, the angle between the curves is preserved under a conformal mapping
• The preservation of angles is a consequence of the Cauchy-Riemann equations, which ensure that the Jacobian matrix of the mapping is a scalar multiple of a rotation matrix

Bilinear transformations

• A bilinear transformation (or Möbius transformation) is a special type of conformal mapping that maps the extended complex plane (the complex plane along with the point at infinity) onto itself
• The general form of a bilinear transformation is: $w = \frac{az + b}{cz + d}$, where $a, b, c, d$ are complex constants satisfying $ad - bc \neq 0$
• Bilinear transformations are used in control theory to map the stability region of a system (e.g., the left half-plane) to a more convenient domain (e.g., the unit disk)
• Common bilinear transformations include the Joukowsky transformation and the Cayley transform

Mapping of regions

• Conformal mapping can be used to map one region in the complex plane to another region with a simpler geometry
• For example, the exterior of a unit disk can be mapped to the upper half-plane using the Joukowsky transformation: $w = \frac{1}{2}(z + \frac{1}{z})$
• The mapping of regions is useful in control theory for analyzing the stability and performance of systems in different domains
• By mapping a system's transfer function to a simpler domain, the analysis and design of the system can be greatly simplified

Laplace transform

• The Laplace transform is an integral transform that converts a time-domain function $f(t)$ into a complex frequency-domain function $F(s)$
• The Laplace transform is defined as: $F(s) = \mathcal{L}\{f(t)\} = \int_0^{\infty} f(t) e^{-st} dt$, where $s$ is a complex variable
• The Laplace transform is a powerful tool in control theory for analyzing and designing linear time-invariant (LTI) systems

Definition and properties

• The Laplace transform has several important properties that make it useful for solving differential equations and analyzing LTI systems:
  • Linearity: $\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)$
  • Time shifting: $\mathcal{L}\{f(t-a)u(t-a)\} = e^{-as}F(s)$, where $u(t)$ is the unit step function
  • Frequency shifting: $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$
  • Differentiation: $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$
  • Integration: $\mathcal{L}\{\int_0^t f(\tau) d\tau\} = \frac{1}{s}F(s)$
• These properties allow for the transformation of differential equations into algebraic equations in the complex frequency domain

Inverse Laplace transform

• The inverse Laplace transform converts a complex frequency-domain function $F(s)$ back into a time-domain function $f(t)$
• The inverse Laplace transform is defined as: $f(t) = \mathcal{L}^{-1}\{F(s)\} = \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma+i\infty} F(s) e^{st} ds$, where $\gamma$ is a real constant greater than the real part of all singularities of $F(s)$
• The inverse Laplace transform can be computed using various techniques, such as partial fraction expansion, the convolution theorem, and the residue theorem
• In control theory, the inverse Laplace transform is used to obtain the time-domain response of a system from its transfer function

Application to differential equations

• The Laplace transform is particularly useful for solving linear differential equations with initial conditions
• By applying the Laplace transform to a differential equation, the equation is converted into an algebraic equation in the complex frequency domain
• The algebraic equation can be solved for the transformed function $F(s)$, and the time-domain solution $f(t)$ can be obtained using the inverse Laplace transform
• This technique is widely used in control theory to analyze the transient and steady-state behavior of LTI systems, as well as to design controllers and filters

Fourier transform

• The Fourier transform is an integral transform that represents a time-domain function $f(t)$ as a continuous spectrum of complex exponentials in the frequency domain $F(\omega)$
• The Fourier transform is defined as: $F(\omega) = \mathcal{F}\{f(t)\} = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$, where $\omega$ is the angular frequency
• The Fourier transform is used in control theory to analyze the frequency response of systems and to design filters and controllers

Definition and properties

• The Fourier transform has several important properties that make it useful for analyzing signals and systems:
  • Linearity: $\mathcal{F}{af(t) + bg(