📐Complex Analysis Unit 9 – Harmonic Functions

Key Concepts and Definitions

• Harmonic functions are real-valued functions that satisfy Laplace's equation $\nabla^2u=0$ in a domain $D\subset\mathbb{R}^n$
• In complex analysis, harmonic functions are the real and imaginary parts of holomorphic (analytic) functions
  • If $f(z)=u(x,y)+iv(x,y)$ is holomorphic, then $u$ and $v$ are harmonic functions
• Harmonic conjugate refers to the function $v$ that satisfies the Cauchy-Riemann equations with a given harmonic function $u$
• Mean value property states that the value of a harmonic function at any point is equal to the average of its values on any sphere or circle centered at that point
• Maximum and minimum principles assert that a non-constant harmonic function cannot attain its maximum or minimum value inside its domain
• Dirichlet problem involves finding a harmonic function that takes prescribed values on the boundary of a domain
• Green's functions are fundamental solutions to Laplace's equation and are used to solve boundary value problems

Properties of Harmonic Functions

• Harmonic functions are infinitely differentiable (smooth) in their domain of definition
• The sum, difference, and scalar multiple of harmonic functions are also harmonic
• Harmonic functions satisfy the mean value property for balls and spheres
  • $u(x_0)=\frac{1}{V(B_r(x_0))}\int_{B_r(x_0)}u(x)dV$ for balls
  • $u(x_0)=\frac{1}{A(S_r(x_0))}\int_{S_r(x_0)}u(x)dA$ for spheres
• The composition of a harmonic function with a conformal mapping is also harmonic
• Harmonic functions are invariant under orthogonal transformations (rotations and reflections)
• The product of two harmonic functions is not necessarily harmonic, but the Poisson bracket of two harmonic functions is harmonic
• Harnack's inequality provides an estimate for the values of a positive harmonic function in terms of its values on a boundary

Laplace's Equation

• Laplace's equation is a second-order partial differential equation given by $\nabla^2u=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=0$
• Solutions to Laplace's equation are called harmonic functions
• Laplace's equation arises in various physical contexts, such as electrostatics, fluid dynamics, and heat conduction
• In two dimensions, Laplace's equation simplifies to $\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0$
• Laplace's equation is an elliptic partial differential equation, which means it has no real characteristic curves
• The Laplace operator $\nabla^2$ is invariant under rotations and translations
• Fundamental solutions to Laplace's equation include the logarithmic potential in 2D and the Newtonian potential in 3D

Connection to Complex Analysis

• Harmonic functions are closely related to holomorphic (analytic) functions in complex analysis
• If $f(z)=u(x,y)+iv(x,y)$ is holomorphic, then $u$ and $v$ are harmonic functions satisfying the Cauchy-Riemann equations
  • $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$
• Given a harmonic function $u$, its harmonic conjugate $v$ can be found (up to a constant) using the Cauchy-Riemann equations
• The real and imaginary parts of a holomorphic function form a harmonic conjugate pair
• Harmonic functions can be used to construct the real and imaginary parts of complex analytic functions
• The maximum modulus principle for holomorphic functions is a consequence of the maximum principle for harmonic functions
• Cauchy's integral formula can be used to express holomorphic functions in terms of their boundary values, analogous to the Poisson integral formula for harmonic functions

Examples of Harmonic Functions

• Constant functions are the simplest examples of harmonic functions
• Linear functions of the form $ax+by+c$ are harmonic in any domain
• The real and imaginary parts of the complex exponential function $e^z=e^x\cos y+ie^x\sin y$ are harmonic
• The real and imaginary parts of the complex logarithm $\log z=\ln|z|+i\arg z$ are harmonic (except at the origin)
• The Poisson kernel $P(r,\theta)=\frac{1-r^2}{1-2r\cos\theta+r^2}$ is a harmonic function in the unit disk
• Green's functions for Laplace's equation, such as $G(x,y)=-\frac{1}{2\pi}\ln|x-y|$ in 2D and $G(x,y)=\frac{1}{4\pi|x-y|}$ in 3D, are harmonic (except at the singularity)
• The real and imaginary parts of the complex polynomial $z^n$ are harmonic functions

Applications in Physics and Engineering

• Harmonic functions appear in various physical contexts where Laplace's equation arises
• In electrostatics, the electric potential in a charge-free region is a harmonic function
  • Solving Laplace's equation with appropriate boundary conditions gives the electric field
• In fluid dynamics, the velocity potential of an irrotational and incompressible flow is a harmonic function
  • Streamlines and equipotential lines form orthogonal families in 2D
• In heat conduction, the steady-state temperature distribution in a region without heat sources or sinks is a harmonic function
• Harmonic functions are used in the study of gravitational and magnetic fields, as well as in elasticity theory
• In engineering, harmonic functions are used to model steady-state heat transfer, fluid flow, and electrostatic fields
• Conformal mappings, which preserve harmonic functions, are used in solving 2D problems with complex geometries

Boundary Value Problems

• Boundary value problems involve finding a harmonic function that satisfies given conditions on the boundary of a domain
• Dirichlet problem seeks a harmonic function with prescribed values on the boundary
  • Existence and uniqueness of solutions depend on the regularity of the boundary and the continuity of the boundary data
• Neumann problem seeks a harmonic function with prescribed normal derivative on the boundary
  • The solution is unique up to an additive constant
• Mixed boundary value problems involve a combination of Dirichlet and Neumann conditions on different parts of the boundary
• Poisson integral formula expresses the solution to the Dirichlet problem in terms of the boundary values
  • $u(re^{i\theta})=\frac{1}{2\pi}\int_0^{2\pi}P(r,\theta-t)f(e^{it})dt$ for the unit disk
• Green's functions are used to solve boundary value problems by reducing them to integral equations
• Conformal mappings can be used to transform boundary value problems to simpler domains

Advanced Topics and Extensions

• Subharmonic and superharmonic functions are generalizations of harmonic functions that satisfy inequalities involving the Laplace operator
  • Subharmonic functions satisfy $\nabla^2u\geq0$, while superharmonic functions satisfy $\nabla^2u\leq0$
• Harmonic functions on Riemannian manifolds are defined using the Laplace-Beltrami operator, which generalizes the Laplace operator to curved spaces
• Discrete harmonic functions are defined on graphs and networks, where the Laplace operator is replaced by the graph Laplacian matrix
• Harmonic measure quantifies the distribution of Brownian motion paths that exit a domain at different boundary points
• Harmonic maps are mappings between Riemannian manifolds that satisfy a generalized Laplace equation and have important applications in geometry and physics
• Potential theory studies the properties of harmonic functions and their generalizations, such as subharmonic and superharmonic functions
• The study of harmonic functions extends to other partial differential equations, such as the heat equation and the wave equation, which have time-dependent solutions