Complex Analysis: Functions and Numbers

• Complex analysis explores functions of complex numbers.
• It applies to diverse fields like algebraic geometry, number theory, analytic combinatorics, applied mathematics, physics, hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and nuclear engineering.

Complex Numbers

• Complex numbers have the form a + bi, where a and b are real numbers, and i equals the square root of -1.
• Geometrically, complex numbers are points on the complex plane; the x-axis is the real part, and the y-axis is the imaginary part.
• Polar form: complex numbers as r(cos θ + i sin θ) or re^(iθ), where r is magnitude/modulus, θ is the argument.
• Euler's formula links complex exponentials to trigonometric functions: e^(iθ) = cos θ + i sin θ.

Complex Functions

• Complex functions map complex numbers to complex numbers.
• They are written as f(z) = u(x, y) + iv(x, y), where z = x + iy and u and v are real-valued functions.

Holomorphic Functions

• A holomorphic (or analytic) function is complex differentiable in a neighborhood of every point in its domain.
• Complex differentiability requires that lim (f(z + h) - f(z))/h exists as h approaches 0, regardless of direction.
• Holomorphic functions are infinitely differentiable with a power series (Taylor series) representation near each domain point.
• Cauchy-Riemann equations: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x (necessary condition for holomorphic functions).
• If partial derivatives of u and v are continuous and satisfy Cauchy-Riemann equations, then f(z) = u(x, y) + iv(x, y) is holomorphic.

Complex Integration

• Complex integration integrates a complex function along a path in the complex plane.
• A path is a continuous function from a real interval to the complex plane.
• The integral of f(z) along path γ is written ∫γ f(z) dz.
• Cauchy's integral theorem: if f(z) is holomorphic in simply connected domain D and γ is a closed path in D, then ∫γ f(z) dz = 0.
• Cauchy's integral formula: if f(z) is holomorphic in simply connected domain D and γ is a closed path in D enclosing point a, then f(a) = (1/2πi) ∫γ f(z)/(z - a) dz.
• Cauchy's integral formula can be used to compute holomorphic function values and derivatives within a domain.

Series Representations

• Taylor series represent holomorphic functions within their convergence radius: f(z) = Σ [f^(n)(a) / n!] * (z - a)^n.
• Laurent series represent functions holomorphic in an annulus: f(z) = Σ a_n (z - a)^n, with both positive and negative powers of (z - a).

Singularities and Residues

• A singularity is a point where a complex function fails to be holomorphic.
• Isolated singularities have a neighborhood devoid of other singularities.
• Types of isolated singularities:
  • Removable singularities: the function's limit exists.
  • Poles: the function approaches infinity.
  • Essential singularities: erratic behavior.
• The residue of f(z) at isolated singularity a is the coefficient a_{-1} of the (z - a)^{-1} term in the Laurent series of f(z) around a.
• The residue theorem: if f(z) is holomorphic in domain D except for isolated singularities a_1, a_2, ..., a_n in D, and γ is a closed path in D enclosing these singularities, then ∫γ f(z) dz = 2πi Σ Res(f, a_k), where Res(f, a_k) is the residue of f at a_k.
• The residue theorem is used for evaluating complex integrals and has applications in physics and engineering.

Applications

• Fluid dynamics problems, such as fluid flow around obstacles.
• Analysis of electrical circuits and electromagnetic fields.
• Quantum mechanics, for studying particle behavior.
• Signal processing, for signal analysis and manipulation.
• Number theory, specifically the distribution of prime numbers.