📐Complex Analysis Unit 2 – Complex Functions and Mappings

Key Concepts and Definitions

• Complex numbers consist of a real part and an imaginary part, denoted as $z = a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($i^2 = -1$)
• The complex plane is a 2D representation of complex numbers, with the real part on the horizontal axis and the imaginary part on the vertical axis
  • Each point in the complex plane corresponds to a unique complex number
  • The distance from the origin to a point $z$ is called the modulus or absolute value of $z$, denoted as $|z|$
• Complex functions map complex numbers from one complex plane (the domain) to another complex plane (the codomain)
  • A complex function $f(z)$ can be written as $f(z) = u(x, y) + iv(x, y)$, where $u$ and $v$ are real-valued functions and $z = x + iy$
• Limit of a complex function $f(z)$ as $z$ approaches a point $z_0$ is defined as $\lim_{z \to z_0} f(z) = L$ if, for any $\varepsilon > 0$, there exists a $\delta > 0$ such that $|f(z) - L| < \varepsilon$ whenever $0 < |z - z_0| < \delta$
• Continuity of a complex function $f(z)$ at a point $z_0$ means that $\lim_{z \to z_0} f(z) = f(z_0)$
• Differentiability of a complex function $f(z)$ at a point $z_0$ means that the limit $\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$ exists, and this limit is called the derivative of $f$ at $z_0$, denoted as $f'(z_0)$

Complex Functions and Their Properties

• Complex functions can be represented by power series, such as the exponential function $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$ and the logarithmic function $\log(1 + z) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{z^n}{n}$
• The composition of two complex functions $f(z)$ and $g(z)$ is defined as $(f \circ g)(z) = f(g(z))$
• The inverse of a complex function $f(z)$, denoted as $f^{-1}(z)$, satisfies $f(f^{-1}(z)) = f^{-1}(f(z)) = z$
  • Not all complex functions have an inverse, and the existence of an inverse depends on the function being one-to-one (injective)
• Complex functions can be even, satisfying $f(-z) = f(z)$ (cosine function), or odd, satisfying $f(-z) = -f(z)$ (sine function)
• Periodic complex functions satisfy $f(z + p) = f(z)$ for some non-zero complex number $p$, called the period (exponential function with imaginary argument)
• Bounded complex functions have an upper limit on their absolute value, i.e., there exists an $M > 0$ such that $|f(z)| \leq M$ for all $z$ in the domain (constant functions)

Analytic Functions and Cauchy-Riemann Equations

• Analytic functions are complex functions that are differentiable at every point in their domain
  • Analyticity is a stronger condition than differentiability, as it requires the function to be differentiable in a neighborhood of each point
• For a complex function $f(z) = u(x, y) + iv(x, y)$ to be analytic, it must satisfy the Cauchy-Riemann equations:
  • $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
  • These equations establish a relationship between the real and imaginary parts of an analytic function
• Harmonic functions are real-valued functions that satisfy Laplace's equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
  • The real and imaginary parts of an analytic function are harmonic functions
• Entire functions are analytic functions that are defined and analytic on the whole complex plane (exponential function, polynomials)
• Meromorphic functions are analytic functions except at a set of isolated points, called poles, where they have singularities (rational functions)
• The Cauchy-Riemann equations can be used to find the analytic continuation of a complex function defined on a limited domain to a larger domain

Mapping Techniques and Transformations

• Complex functions can be visualized as mappings between two complex planes, transforming shapes and regions from the domain to the codomain
• Linear transformations of the form $f(z) = az + b$, where $a, b \in \mathbb{C}$ and $a \neq 0$, represent rotation, scaling, and translation in the complex plane
  • The multiplication by $a$ rotates and scales the plane, while the addition of $b$ translates the plane
• Möbius transformations, also known as linear fractional transformations, are of the form $f(z) = \frac{az + b}{cz + d}$, where $a, b, c, d \in \mathbb{C}$ and $ad - bc \neq 0$
  • Möbius transformations map circles and lines to circles and lines, preserving angles (conformal mappings)
  • They are used in modeling physical systems and solving problems in geometry and physics
• The exponential function $e^z$ maps horizontal lines to circles centered at the origin and vertical lines to radial lines emanating from the origin
• The logarithmic function $\log(z)$ maps the right half-plane to horizontal strips and the unit circle to the imaginary axis
• The power function $z^n$ maps the complex plane to itself, wrapping it around the origin $n$ times
  • For fractional powers, the function is multi-valued, requiring branch cuts to maintain single-valuedness

Conformal Mappings and Applications

• Conformal mappings are complex functions that preserve angles between curves at each point
  • Analytic functions with non-zero derivatives are conformal mappings
• Conformal mappings have many applications in physics and engineering, such as:
  • Fluid dynamics: Mapping the flow around objects to simpler geometries
  • Electrostatics: Mapping the electric field around conductors to simpler geometries
  • Heat transfer: Mapping the temperature distribution in irregular shapes to simpler geometries
• The Joukowsky transformation, given by $f(z) = \frac{1}{2}(z + \frac{1}{z})$, maps circles to airfoil shapes, which is useful in aerodynamics
• The Schwarz-Christoffel transformation maps the upper half-plane to polygonal regions, which is useful in solving boundary value problems
• Conformal mappings can be used to solve Laplace's equation in two dimensions by mapping the problem domain to a simpler domain where the solution is known
  • The solution in the original domain is then obtained by applying the inverse mapping
• Conformal mappings preserve harmonic functions, meaning that if $u(x, y)$ is harmonic, then $u(f(z))$ is also harmonic, where $f(z)$ is a conformal mapping

Singularities and Residues

• Singularities are points where a complex function is not analytic, and they can be classified into three types:
  • Removable singularities: The limit of the function exists at the point, and the function can be redefined to be analytic there (e.g., $f(z) = \frac{\sin(z)}{z}$ at $z = 0$)
  • Poles: The limit of the function does not exist, but the limit of the product of the function and a power of $(z - z_0)$ does exist (e.g., $f(z) = \frac{1}{z}$ at $z = 0$)
  • Essential singularities: The limit of the function does not exist, and the function behaves erratically near the point (e.g., $f(z) = e^{\frac{1}{z}}$ at $z = 0$)
• The residue of a 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 $z_0$
  • For a simple pole, the residue is given by $\lim_{z \to z_0} (z - z_0)f(z)$
  • For a pole of order $n$, the residue is given by $\frac{1}{(n-1)!} \lim_{z \to z_0} \frac{d^{n-1}}{dz^{n-1}} ((z - z_0)^n f(z))$
• The residue theorem states that the integral of a function $f(z)$ along a closed contour $C$ is equal to $2\pi i$ times the sum of the residues of $f(z)$ at the poles enclosed by $C$
  • This theorem is useful for evaluating integrals of complex functions along closed contours
• The argument principle relates the number of zeros and poles of a meromorphic function inside a closed contour to the change in the argument of the function along the contour
• The Mittag-Leffler theorem states that any meromorphic function can be expressed as a sum of its principal part at each pole plus an entire function

Integration in the Complex Plane

• Integration of complex functions is performed using contour integrals, which are integrals along a curve in the complex plane
  • The contour integral of a function $f(z)$ along a curve $C$ is defined as $\int_C f(z) dz$
• Cauchy's integral theorem states that if $f(z)$ is analytic in a simply connected domain $D$ and $C$ is a closed contour lying entirely in $D$, then $\int_C f(z) dz = 0$
  • This theorem is the foundation for many important results in complex analysis
• Cauchy's integral formula states that if $f(z)$ is analytic in a simply connected domain $D$ and $z_0$ is a point in $D$, then $f(z_0) = \frac{1}{2\pi i} \int_C \frac{f(z)}{z - z_0} dz$, where $C$ is a closed contour enclosing $z_0$
  • This formula allows the calculation of the value of an analytic function at a point using a contour integral
• The residue theorem can be used to evaluate real integrals by extending the integrand to the complex plane and choosing an appropriate contour
  • This technique is particularly useful for evaluating improper integrals and integrals involving trigonometric or exponential functions
• The Cauchy principal value of an improper integral is a method for assigning a finite value to an integral that would otherwise be undefined due to a singularity on the path of integration
• The Jordan curve theorem states that a simple closed curve divides the complex plane into two distinct regions: the interior and the exterior
  • This theorem is important for understanding the behavior of complex functions and their integrals

Practical Applications and Examples

• Complex analysis has numerous applications in various fields, such as:
  • Physics: Quantum mechanics, electromagnetism, fluid dynamics
  • Engineering: Signal processing, control theory, electrical networks
  • Mathematics: Number theory, algebraic geometry, differential equations
• The Fourier transform, which is widely used in signal processing, can be viewed as a complex function that maps a time-domain signal to a frequency-domain representation
  • The inverse Fourier transform maps the frequency-domain representation back to the time-domain signal
• The Laplace transform, used in solving differential equations and analyzing control systems, is a complex function that maps a time-domain function to a complex frequency-domain representation
  • The inverse Laplace transform maps the complex frequency-domain representation back to the time-domain function
• The Gamma function, which is an extension of the factorial function to complex numbers, has applications in probability theory, combinatorics, and number theory
  • The Gamma function satisfies the functional equation $\Gamma(z+1) = z\Gamma(z)$ and has poles at the non-positive integers
• The Riemann zeta function, which is defined as $\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$ for $\Re(s) > 1$, has deep connections to prime numbers and the distribution of primes
  • The Riemann hypothesis, which states that the non-trivial zeros of the zeta function have real part equal to $\frac{1}{2}$, is one of the most important unsolved problems in mathematics
• Conformal mappings are used in the design of airfoils and other aerodynamic shapes, as they allow the flow around the object to be studied in a simpler geometric setting
  • The Kutta-Joukowski theorem relates the lift generated by an airfoil to the circulation of the flow around it, which can be calculated using complex analysis techniques