💠Intro to Complex Analysis Unit 3 – Elementary Functions in Complex Analysis

Key Concepts and Definitions

• Complex numbers extend the real number system by introducing the imaginary unit $i$ where $i^2 = -1$
• A complex number $z$ is written in the form $z = a + bi$ where $a$ and $b$ are real numbers and $i$ is the imaginary unit
• The real part of $z$ is denoted as $\text{Re}(z) = a$ and the imaginary part is denoted as $\text{Im}(z) = b$
• The complex conjugate of $z = a + bi$ is defined as $\bar{z} = a - bi$
• The modulus or absolute value of a complex number $z$ is defined as $|z| = \sqrt{a^2 + b^2}$
• The argument or phase of a complex number $z$ is defined as $\arg(z) = \arctan(\frac{b}{a})$
• Euler's formula establishes the relationship between complex exponentials and trigonometric functions: $e^{ix} = \cos(x) + i\sin(x)$

Complex Numbers and the Complex Plane

• The complex plane is a 2D representation of complex numbers where the horizontal axis represents the real part and the vertical axis represents the imaginary part
• A complex number $z = a + bi$ can be plotted on the complex plane as a point $(a, b)$
• The distance from the origin to the point $(a, b)$ represents the modulus of the complex number $|z|$
• The angle formed by the positive real axis and the line connecting the origin to the point $(a, b)$ represents the argument of the complex number $\arg(z)$
• Complex numbers can be represented in various forms:
  • Rectangular or Cartesian form: $z = a + bi$
  • Polar form: $z = r(\cos(\theta) + i\sin(\theta))$ where $r = |z|$ and $\theta = \arg(z)$
  • Exponential form: $z = re^{i\theta}$ using Euler's formula
• Addition and subtraction of complex numbers are performed component-wise: $(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$
• Multiplication of complex numbers follows the distributive law and the property $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Elementary Functions in the Complex Domain

• Elementary functions such as exponential, logarithmic, and trigonometric functions can be extended to the complex domain
• The complex exponential function is defined as $e^z = e^{a+bi} = e^a(\cos(b) + i\sin(b))$
• The complex logarithm is defined as the inverse of the complex exponential function: $\log(z) = \ln|z| + i\arg(z)$
  • The complex logarithm is multi-valued due to the periodicity of the argument
• The complex trigonometric functions (sine, cosine, and tangent) can be defined using Euler's formula:
  • $\sin(z) = \frac{e^{iz} - e^{-iz}}{2i}$
  • $\cos(z) = \frac{e^{iz} + e^{-iz}}{2}$
  • $\tan(z) = \frac{\sin(z)}{\cos(z)}$
• The complex hyperbolic functions (sinh, cosh, and tanh) are defined similarly using exponentials:
  • $\sinh(z) = \frac{e^z - e^{-z}}{2}$
  • $\cosh(z) = \frac{e^z + e^{-z}}{2}$
  • $\tanh(z) = \frac{\sinh(z)}{\cosh(z)}$
• The complex power function $z^w$ is defined using the complex exponential and logarithm: $z^w = e^{w\log(z)}$

Properties of Complex Functions

• A complex function $f(z)$ maps complex numbers from one complex plane (the z-plane) to another complex plane (the w-plane)
• The domain of a complex function is the set of complex numbers for which the function is defined
• The range of a complex function is the set of complex numbers that the function can output
• A complex function is continuous at a point $z_0$ if $\lim_{z \to z_0} f(z) = f(z_0)$
  • Continuity in the complex domain is similar to continuity in the real domain
• A complex function is differentiable at a point $z_0$ if the limit $\lim_{h \to 0} \frac{f(z_0 + h) - f(z_0)}{h}$ exists
  • The derivative of a complex function $f(z)$ is denoted as $f'(z)$ or $\frac{df}{dz}$
• The Cauchy-Riemann equations provide a necessary condition for a complex function to be differentiable:
  • If $f(z) = u(x, y) + iv(x, y)$ is differentiable, then $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
• A complex function is analytic or holomorphic if it is differentiable at every point in its domain

Analytic Functions and Differentiability

• Analytic functions have several important properties:
  • They are infinitely differentiable
  • They can be represented by power series expansions
  • They satisfy the Cauchy-Riemann equations
• The sum, difference, product, and quotient of analytic functions are also analytic
• The composition of analytic functions is analytic
• If a complex function is analytic, its real and imaginary parts are harmonic functions, satisfying Laplace's equation: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ and $\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$
• The Cauchy Integral Formula relates the values of an analytic function inside a region to its values on the boundary:
  • If $f(z)$ is analytic inside and on a simple closed curve $C$, then $f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} dz$ for any point $z_0$ inside $C$
• Morera's Theorem provides a sufficient condition for a complex function to be analytic:
  • If $f(z)$ is continuous in a domain $D$ and $\oint_C f(z) dz = 0$ for every closed curve $C$ in $D$, then $f(z)$ is analytic in $D$

Singularities and Special Points

• A singularity of a complex function is a point where the function is not analytic
• Types of singularities:
  • Removable singularity: The function can be redefined at the point to make it analytic (e.g., $f(z) = \frac{\sin(z)}{z}$ at $z = 0$)
  • Pole: The function tends to infinity as the point is approached (e.g., $f(z) = \frac{1}{z}$ at $z = 0$)
  • Essential singularity: The function exhibits complex behavior near the point (e.g., $f(z) = e^{\frac{1}{z}}$ at $z = 0$)
• The residue of a complex function at a singularity is the coefficient of the $\frac{1}{z}$ term in its Laurent series expansion
• The Residue Theorem relates the residues of a function inside a region to the integral of the function along the boundary:
  • If $f(z)$ is analytic inside and on a simple closed curve $C$ except for a finite number of singularities inside $C$, then $\oint_C f(z) dz = 2\pi i \sum_{k=1}^n \text{Res}(f, z_k)$ where $z_k$ are the singularities inside $C$
• Branch points and branch cuts are related to multi-valued functions (e.g., logarithm and power functions)
  • A branch point is a point where the function is multi-valued
  • A branch cut is a curve connecting branch points, used to define a single-valued branch of the function

Applications and Examples

• Complex analysis has numerous applications in various fields, including:
  • Physics (e.g., quantum mechanics, fluid dynamics, electromagnetism)
  • Engineering (e.g., signal processing, control theory, electrical networks)
  • Mathematics (e.g., number theory, geometry, topology)
• Example: Evaluating real integrals using contour integration
  • Some real integrals can be more easily evaluated by converting them to complex contour integrals and using the Residue Theorem
  • For instance, $\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$ can be evaluated by considering the contour integral of $\frac{e^{iz}}{z^2 + 1}$ along a semicircular contour in the upper half-plane
• Example: Solving Laplace's equation in 2D using analytic functions
  • Laplace's equation $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$ arises in many physical problems (e.g., heat conduction, electrostatics)
  • Solutions to Laplace's equation can be found using analytic functions, as the real and imaginary parts of an analytic function satisfy Laplace's equation
• Example: Modeling fluid flow using complex potential functions
  • In 2D ideal fluid flow, the velocity field can be represented by the gradient of a complex potential function $\phi(z) = \varphi(x, y) + i\psi(x, y)$
  • The real part $\varphi(x, y)$ is the velocity potential, and the imaginary part $\psi(x, y)$ is the stream function
  • Complex analysis techniques can be used to find the complex potential function for various flow scenarios (e.g., flow around a cylinder, flow in a corner)

Common Pitfalls and Tips

• Remember that the complex logarithm and power functions are multi-valued
  • When working with these functions, be careful to choose the appropriate branch and maintain consistency
• Be cautious when applying the Cauchy Integral Formula or the Residue Theorem
  • Ensure that the function satisfies the necessary conditions (e.g., analyticity, singularities) and that the contour is chosen appropriately
• When evaluating contour integrals, pay attention to the direction of the contour and the orientation of the complex plane
  • The direction of the contour affects the sign of the integral
  • The orientation of the complex plane (counterclockwise or clockwise) affects the sign of the residues
• When using the Cauchy-Riemann equations to check for differentiability, remember that they are necessary but not sufficient conditions
  • A function may satisfy the Cauchy-Riemann equations but still not be differentiable (e.g., $f(z) = |z|^2$ at $z = 0$)
• When working with series expansions (e.g., Taylor series, Laurent series), be aware of the region of convergence
  • The region of convergence determines where the series representation is valid and can be used for analysis
• Practice visualizing complex functions and their properties using graphical tools
  • Plotting the real and imaginary parts, the modulus, and the argument of a complex function can provide valuable insights
  • Software tools like MATLAB, Mathematica, or Python libraries (e.g., NumPy, SciPy) can be helpful for visualization and computation