Complex functions are the building blocks of complex analysis. They map complex numbers to other complex numbers, often represented as f(z) = u(x, y) + iv(x, y). Understanding their properties is key to grasping more advanced concepts.

Continuity, differentiability, and analyticity are crucial properties of complex functions. The Cauchy-Riemann equations provide a powerful tool for determining analyticity, while harmonic functions connect complex analysis to real-world applications like fluid dynamics and electrostatics.

Complex Functions and Domains

Defining Complex Functions

• A complex function maps complex numbers from one set (the domain) to another set (the codomain) of complex numbers
• Complex functions can be represented in the form $f(z) = u(x, y) + iv(x, y)$, where $z = x + iy$ and $u, v$ are real-valued functions
• The real part of a complex function is denoted by $Re(f(z)) = u(x, y)$, and the imaginary part is denoted by $Im(f(z)) = v(x, y)$
• Examples of complex functions include polynomial functions ($f(z) = z^2 + 3z - 1$), exponential functions ($f(z) = e^z$), trigonometric functions ($f(z) = \sin(z)$), and logarithmic functions ($f(z) = \log(z)$)

Domains of Complex Functions

• The domain of a complex function is the set of all complex numbers for which the function is defined and produces a unique output value
• The domain can be represented as a subset of the complex plane
• Functions may have restricted domains due to the presence of singularities or branch cuts
• For example, the domain of the logarithmic function $f(z) = \log(z)$ is the set of all non-zero complex numbers, as the logarithm is not defined for zero

Continuity, Differentiability, and Analyticity

Continuity of Complex Functions

• A complex function $f(z)$ is continuous at a point $z_0$ if and only if the limit of $f(z)$ as $z$ approaches $z_0$ exists and equals $f(z_0)$
• Continuity of complex functions is similar to that of real-valued functions
• The sum, difference, product, and quotient of continuous functions are also continuous
• If a function is continuous on a closed and bounded domain, it is uniformly continuous on that domain

Differentiability and Analyticity

• A complex function $f(z)$ is differentiable at a point $z_0$ if and only if the limit of $(f(z) - f(z_0)) / (z - z_0)$ exists as $z$ approaches $z_0$. This limit is called the derivative of $f$ at $z_0$ and is denoted by $f'(z_0)$
• A complex function $f(z)$ is analytic (or holomorphic) on a domain $D$ if it is differentiable at every point in $D$
• If a complex function is analytic, it is infinitely differentiable, and its Taylor series converges to the function in a neighborhood of every point in its domain
• The sum, difference, product, quotient, and composition of analytic functions are also analytic
• Examples of analytic functions include polynomials, exponential functions, and trigonometric functions

Cauchy-Riemann Equations for Analyticity

Stating the Cauchy-Riemann Equations

• The Cauchy-Riemann equations provide a necessary and sufficient condition for a complex function $f(z) = u(x, y) + iv(x, y)$ to be analytic
• The Cauchy-Riemann equations are: $\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$, where $\partial$ denotes the partial derivative
• If the partial derivatives of $u$ and $v$ exist and are continuous, and the Cauchy-Riemann equations are satisfied at a point $z_0$, then $f(z)$ is analytic at $z_0$

Applying the Cauchy-Riemann Equations

• If $f(z)$ is analytic in a domain $D$, then the Cauchy-Riemann equations are satisfied at every point in $D$
• The Cauchy-Riemann equations can be used to determine the analyticity of complex functions and to find the derivative of an analytic function
• To check if a function is analytic, compute the partial derivatives of its real and imaginary parts and verify that they satisfy the Cauchy-Riemann equations
• For example, consider the function $f(z) = z^2 = (x + iy)^2 = (x^2 - y^2) + i(2xy)$. The partial derivatives are $\frac{\partial u}{\partial x} = 2x$, $\frac{\partial v}{\partial y} = 2x$, $\frac{\partial u}{\partial y} = -2y$, and $\frac{\partial v}{\partial x} = 2y$. Since these satisfy the Cauchy-Riemann equations, $f(z)$ is analytic

Harmonic Functions vs Analytic Functions

Properties of Harmonic Functions

• A real-valued function $h(x, y)$ is harmonic if it satisfies Laplace's equation: $\frac{\partial^2 h}{\partial x^2} + \frac{\partial^2 h}{\partial y^2} = 0$
• Harmonic functions have several important properties, such as the mean value property, the maximum principle, and the uniqueness principle
• The mean value property states that the value of a harmonic function at any point is equal to the average of its values on any circle centered at that point
• The maximum principle states that a non-constant harmonic function cannot attain its maximum or minimum value inside its domain
• The uniqueness principle states that if two harmonic functions have the same boundary values on a bounded domain, they are identical throughout the domain

Relationship between Harmonic and Analytic 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
• Conversely, if $u(x, y)$ and $v(x, y)$ are harmonic functions that satisfy the Cauchy-Riemann equations, then $f(z) = u(x, y) + iv(x, y)$ is an analytic function
• The real and imaginary parts of an analytic function are related by the Cauchy-Riemann equations and form a harmonic conjugate pair
• Harmonic functions play a crucial role in various applications of complex analysis, such as fluid dynamics (potential flow), electrostatics (electric potential), and heat conduction (temperature distribution)