💠Intro to Complex Analysis Unit 2 – Analytic Functions in Complex Analysis

Key Concepts and Definitions

• Complex numbers consist of a real part and an imaginary part in the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit defined as $i^2 = -1$
• Complex plane is a 2D representation of complex numbers with the real part on the x-axis and the imaginary part on the y-axis
• Complex functions map complex numbers from one complex plane (domain) to another complex plane (codomain)
  • Example: $f(z) = z^2$ maps complex numbers to their squared values
• Limit of a complex function $f(z)$ as $z$ approaches $z_0$ is defined as $\lim_{z \to z_0} f(z) = L$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $|f(z) - L| < \epsilon$ 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, where $h$ is a complex number approaching zero
• Analytic functions are complex functions that are differentiable at every point in their domain

Complex Differentiability and Analyticity

• Complex differentiability is a stronger condition than real differentiability, as it requires the existence of the limit $\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$ for any complex $h$ approaching zero
• If a complex function $f(z)$ is differentiable at a point $z_0$, then it is also continuous at $z_0$
• Analyticity is an even stronger condition than complex differentiability, requiring the function to be differentiable at every point in its domain
• Analytic functions have many useful properties, such as being infinitely differentiable and satisfying the Cauchy-Riemann equations
• Examples of analytic functions include polynomials ($z^2 + 3z - 1$), exponential functions ($e^z$), and trigonometric functions ($\sin(z)$, $\cos(z)$)
• Non-analytic functions, such as $\bar{z}$ (complex conjugate) and $|z|$ (absolute value), are not differentiable at some or all points in their domain
• Analyticity is preserved under arithmetic operations (addition, subtraction, multiplication, and division) and composition of functions

Cauchy-Riemann Equations

• Cauchy-Riemann equations provide a necessary and sufficient condition for a complex function $f(z) = u(x, y) + iv(x, y)$ to be analytic
• For $f(z)$ to be analytic, the partial derivatives of $u$ and $v$ must satisfy:
  • $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$
  • $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• These equations establish a strong connection between the real and imaginary parts of an analytic function
• Cauchy-Riemann equations can be used to prove that a function is analytic by verifying that the partial derivatives satisfy the equations
• Example: For $f(z) = z^2 = (x + iy)^2 = (x^2 - y^2) + i(2xy)$, we have $u(x, y) = x^2 - y^2$ and $v(x, y) = 2xy$. The partial derivatives satisfy the Cauchy-Riemann equations, confirming that $f(z)$ is analytic
• Cauchy-Riemann equations can also be expressed in polar form for functions given in polar coordinates
• Failure to satisfy the Cauchy-Riemann equations at a point implies that the function is not analytic at that point

Properties of Analytic Functions

• Analytic functions are infinitely differentiable, meaning that all higher-order derivatives exist and are also analytic
• Sum, difference, product, and quotient of two analytic functions are also analytic (as long as the denominator in the quotient is non-zero)
• Composition of two analytic functions is analytic
• Chain rule holds for analytic functions: If $f(z)$ and $g(z)$ are analytic, then $(f \circ g)'(z) = f'(g(z)) \cdot g'(z)$
• Analytic functions satisfy the mean value property: If $f(z)$ is analytic in a disk centered at $z_0$ with radius $r$, then $f(z_0) = \frac{1}{2\pi} \int_0^{2\pi} f(z_0 + re^{i\theta}) d\theta$
• Maximum modulus principle states that if $f(z)$ is analytic and non-constant in a domain $D$, then $|f(z)|$ cannot have a local maximum inside $D$
• Liouville's theorem: If $f(z)$ is analytic and bounded in the entire complex plane, then $f(z)$ must be a constant
• Fundamental theorem of algebra: Every non-constant polynomial with complex coefficients has at least one complex root

Power Series and Taylor Expansions

• Power series is an infinite series of the form $\sum_{n=0}^{\infty} a_n (z - z_0)^n$, where $a_n$ are complex coefficients and $z_0$ is the center of the series
• Convergence of a power series depends on the values of $z$; the series may converge for some $z$ and diverge for others
• Radius of convergence $R$ is the largest radius of a disk centered at $z_0$ within which the power series converges
  • Inside the disk $(|z - z_0| < R)$, the series converges absolutely
  • Outside the disk $(|z - z_0| > R)$, the series diverges
  • On the boundary $(|z - z_0| = R)$, the series may converge conditionally or diverge
• Taylor series is a power series expansion of a function $f(z)$ around a point $z_0$, given by $f(z) = \sum_{n=0}^{\infty} \frac{f^{(n)}(z_0)}{n!} (z - z_0)^n$, where $f^{(n)}(z_0)$ denotes the $n$-th derivative of $f$ at $z_0$
• If a function $f(z)$ is analytic in a disk centered at $z_0$, then it can be represented by its Taylor series within that disk
• Maclaurin series is a special case of the Taylor series when $z_0 = 0$
• Examples of Taylor series expansions:
  • $e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}$
  • $\sin(z) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} z^{2n+1}$
  • $\cos(z) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} z^{2n}$

Singularities and Classification

• Singularities are points in the complex plane where a function is not analytic
• Isolated singularities are singularities that are not surrounded by other singularities in a neighborhood
• Types of isolated singularities:
  • Removable singularity: $\lim_{z \to z_0} (z - z_0) f(z)$ exists and is finite
  • Pole: $\lim_{z \to z_0} (z - z_0) f(z) = \infty$ (of order $m$ if $\lim_{z \to z_0} (z - z_0)^m f(z)$ is finite and non-zero)
  • Essential singularity: $\lim_{z \to z_0} (z - z_0) f(z)$ does not exist or is infinite
• Residue of a function $f(z)$ at an isolated singularity $z_0$ is defined as $\frac{1}{2\pi i} \oint_C f(z) dz$, where $C$ is a closed contour enclosing $z_0$ and no other singularities
• Residue theorem: If $f(z)$ is analytic inside and on a simple closed contour $C$ except for a finite number of isolated singularities $z_1, z_2, \ldots, z_n$ inside $C$, then $\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$
• Branch points are singularities that arise from multi-valued functions, such as $\sqrt{z}$ or $\log(z)$
• Branch cuts are curves in the complex plane used to define a single-valued branch of a multi-valued function

Applications in Physics and Engineering

• Fluid dynamics: Complex potential functions can be used to model 2D incompressible and irrotational fluid flow
  • Velocity potential $\phi(x, y)$ and stream function $\psi(x, y)$ form an analytic function $f(z) = \phi(x, y) + i\psi(x, y)$
  • Streamlines and equipotential lines are perpendicular to each other
• Electrostatics: Electric potential $V(x, y)$ and electric field components $E_x(x, y)$ and $E_y(x, y)$ can be related using analytic functions
  • $E_x = -\frac{\partial V}{\partial x}$ and $E_y = -\frac{\partial V}{\partial y}$ satisfy the Cauchy-Riemann equations
• Quantum mechanics: Wave functions in the position representation can be treated as complex-valued functions
  • Schrödinger equation involves complex-valued wave functions and their derivatives
  • Expectation values of observables are calculated using integrals of complex-valued functions
• Signal processing: Fourier and Laplace transforms use complex exponentials to analyze and manipulate signals
  • Fourier transform maps a time-domain signal to its frequency-domain representation using complex exponentials
  • Laplace transform is a generalization of the Fourier transform that includes complex frequencies and can handle initial conditions
• Control systems: Transfer functions and stability analysis often involve complex variables
  • Transfer functions relate the input and output of a linear time-invariant system in the complex frequency domain
  • Poles and zeros of a transfer function determine the stability and transient response of the system

Common Pitfalls and Tips

• Ensure that the Cauchy-Riemann equations are satisfied at every point in the domain when proving analyticity
• Be cautious when dealing with multi-valued functions, such as logarithms and square roots, as they may have branch cuts and require careful handling
• When evaluating integrals using the residue theorem, make sure to include all singularities inside the contour and use the correct residue formula for each type of singularity
• Remember that the convergence of a power series depends on the value of $z$; always check the radius of convergence and the behavior on the boundary
• When working with Taylor series expansions, be aware of the region of convergence and the accuracy of the approximation within that region
• In applications, pay attention to the physical interpretation of the real and imaginary parts of complex functions, as they often represent different physical quantities
• Practice complex arithmetic, especially multiplication and division, to develop fluency in manipulating complex expressions
• Visualize complex functions and their properties using graphical tools, such as domain coloring or phase portraits, to gain intuition about their behavior