Komplekse Getalle en Funksies

• This document discusses complex numbers and functions, key concepts in mathematical analysis.
• Explores the properties of complex numbers, including their representation, algebra, and geometry.
• Introduces complex functions, focusing on their analyticity, singularities, and applications.

Complex Numbers

• Complex numbers are expressed as z = x + iy, where x and y are real numbers, and i is the imaginary unit (i² = -1).
• The real part of z is x (Re(z)), and the imaginary part is y (Im(z)).
• Complex numbers can be visualized on the complex plane (Argand diagram), with the x-axis representing the real part and the y-axis representing the imaginary part.
• Complex numbers can also be expressed in polar form: z = r(cos θ + i sin θ) = re^(iθ), where r is the magnitude (or modulus) of z, and θ is the argument (or phase) of z.
• The magnitude r is calculated as r = √(x² + y²), and the argument θ is such that cos θ = x/r and sin θ = y/r.
• Euler's formula establishes the relationship between exponential and trigonometric functions: e^(iθ) = cos θ + i sin θ.
• Complex conjugate of z = x + iy is z̄ = x - iy. In polar form, if z = re^(iθ), then z̄ = re^(-iθ)
• Properties: Re(z) = (z + z̄)/2, Im(z) = (z - z̄)/(2i), |z|² = z * z̄.
• Arithmetic operations: addition, subtraction, multiplication, and division are defined for complex numbers.
• For z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂, z₁ + z₂ = (x₁ + x₂) + i(y₁ + y₂), z₁ - z₂ = (x₁ - x₂) + i(y₁ - y₂), z₁ * z₂ = (x₁x₂ - y₁y₂) + i(x₁y₂ + x₂y₁), and z₁/z₂ = [(x₁x₂ + y₁y₂) + i(x₂y₁ - x₁y₂)] / (x₂² + y₂²) provided z₂ ≠ 0.
• De Moivre's Theorem: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ), which simplifies raising a complex number to a power.
• Finding roots of complex numbers: For z = re^(iθ), the nth roots are given by w_k = r^(1/n) * exp[i(θ + 2πk)/n], for k = 0, 1, ..., n-1. There are n distinct nth roots.

Complex Functions

• A complex function is a function that maps complex numbers to complex numbers: w = f(z), where z and w are complex. Can be expressed as w = u(x,y) + iv(x,y), where u and v are real-valued functions of two real variables.
• Limit of a complex function: lim (z→z₀) f(z) = w₀ exists if, for every ε > 0, there exists a δ > 0 such that |f(z) - w₀| < ε whenever 0 < |z - z₀| < δ. The limit must exist regardless of the path taken to approach z₀.
• Continuity: f(z) is continuous at z₀ if lim (z→z₀) f(z) = f(z₀). Both the real and imaginary parts of f(z) must be continuous.
• A complex function f(z) = u(x, y) + iv(x, y) is differentiable at a point z₀ if the limit lim (Δz→0) [f(z₀ + Δz) - f(z₀)] / Δz exists.
• Analyticity: A function f(z) is analytic in a domain D if it is differentiable at every point in D. Analyticity is a stronger condition than differentiability at a single point.
• If f(z) = u(x, y) + iv(x, y) is analytic in a domain D, then the Cauchy-Riemann equations hold: ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x.
• Conversely, if the partial derivatives are continuous and satisfy the Cauchy-Riemann equations, then the function is analytic.
• Harmonic functions: If f(z) = u(x, y) + iv(x, y) is analytic, then both u and v satisfy Laplace's equation: ∂²u/∂x² + ∂²u/∂y² = 0 and ∂²v/∂x² + ∂²v/∂y² = 0. Solutions to Laplace's equation are called harmonic functions.
• u and v are harmonic conjugates of each other. Given u, one can find v (up to a constant) using the Cauchy-Riemann equations.
• Elementary complex functions include polynomials, rational functions, exponential functions, trigonometric functions, hyperbolic functions, and logarithmic functions.
• Complex exponential function: e^z = e^(x+iy) = e^x(cos y + i sin y). It is periodic with period 2πi: e^(z+2πi) = e^z.
• Complex trigonometric functions: sin z = (e^(iz) - e^(-iz))/(2i), cos z = (e^(iz) + e^(-iz))/2, tan z = sin z / cos z, cot z = cos z / sin z, sec z = 1 / cos z, csc z = 1 / sin z.
• Complex hyperbolic functions: sinh z = (e^z - e^(-z))/2, cosh z = (e^z + e^(-z))/2, tanh z = sinh z / cosh z, coth z = cosh z / sinh z, sech z = 1 / cosh z, csch z = 1 / sinh z.
• Complex logarithm: w = ln z is defined as the inverse of the exponential function, z = e^w. If z = re^(iθ), then w = ln r + i(θ + 2nπ), where n is an integer. The complex logarithm is a multi-valued function due to the periodicity of the complex exponential function.
• Principal value of the logarithm: Ln z = ln r + iΘ, where Θ is the principal argument of z, i.e., -π < Θ ≤ π.

Complex Integration

• Complex integration involves integrating a complex function along a path (curve) in the complex plane.
• The integral of f(z) along a curve C parameterized by z(t), a ≤ t ≤ b, is defined as ∫_C f(z) dz = ∫_a^b f(z(t)) z'(t) dt.
• Properties of complex integrals: linearity, additivity over subintervals.
• Cauchy's Theorem: If f(z) is analytic in a simply connected domain D and C is a closed contour in D, then ∫_C f(z) dz = 0.
• Cauchy's Integral Formula: If f(z) is analytic in a simply connected domain D, C is a closed contour in D, and z₀ is a point inside C, then f(z₀) = (1/(2πi)) ∫_C f(z)/(z - z₀) dz.
• Cauchy's Integral Formula for derivatives: f^(n)(z₀) = (n!/(2πi)) ∫_C f(z)/(z - z₀)^(n+1) dz. This formula provides a way to compute the derivatives of an analytic function at a point using a contour integral.

Series

• A complex sequence {z_n} converges to z if for every ε > 0, there exists an N such that |z_n - z| < ε for all n > N.
• A complex series ∑ (from n=1 to ∞) z_n converges to S if the sequence of partial sums S_N = ∑ (from n=1 to N) z_n converges to S.
• Power Series: A power series about a point z₀ is an infinite series of the form ∑ (from n=0 to ∞) a_n (z - z₀)^n, where a_n are complex coefficients.
• Radius of Convergence: For every power series, there exists a number R (0 ≤ R ≤ ∞) such that the series converges absolutely for |z - z₀| < R and diverges for |z - z₀| > R.
• The value R is called the radius of convergence. Several tests can be used to determine the radius of convergence, such as the ratio test and the root test. if lim (n→∞) |a_(n+1) / a_n| = L, then R = 1/L.
• Taylor Series: If f(z) is analytic in a domain D containing z₀, then f(z) can be represented by a Taylor series about z₀: f(z) = ∑ (from n=0 to ∞) [f^(n)(z₀) / n!] (z - z₀)^n.
• Maclaurin Series: A Taylor series about z₀ = 0 is called a Maclaurin series: f(z) = ∑ (from n=0 to ∞) [f^(n)(0) / n!] z^n.
• Laurent Series: If f(z) is analytic in an annulus r < |z - z₀| < R, then f(z) can be represented by a Laurent series: f(z) = ∑ (from n=-∞ to ∞) a_n (z - z₀)^n, where the coefficients a_n are given by a_n = (1/(2πi)) ∫_C f(z) / (z - z₀)^(n+1) dz and C is a closed contour in the annulus.

Singularities and Residues

• A singularity of a complex function f(z) is a point z₀ where f(z) is not analytic.
• Isolated singularity: A singularity z₀ is isolated if there exists a neighborhood of z₀ containing no other singularities.
• Types of Isolated Singularities: Removable singularity, pole, essential singularity.
• Removable singularity: If lim (z→z₀) f(z) exists, then z₀ is a removable singularity. The singularity can be "removed" by defining f(z₀) to be equal to the limit.
• Pole: If lim (z→z₀) (z - z₀)^n f(z) exists and is nonzero for some positive integer n, then z₀ is a pole of order n.
• Essential singularity: If z₀ is neither a removable singularity nor a pole, then it is an essential singularity.
• Residue: The residue of f(z) at an isolated singularity z₀, denoted Res(f, z₀), is the coefficient a₋₁ of the (z - z₀)^(-1) term in the Laurent series expansion of f(z) about z₀.
• If z₀ is a simple pole (pole of order 1), then Res(f, z₀) = lim (z→z₀) (z - z₀) f(z).
• If z₀ is a pole of order n, then Res(f, z₀) = (1/(n-1)!) lim (z→z₀) d^(n-1)/dz^(n-1) [(z - z₀)^n f(z)].
• Residue Theorem: If f(z) is analytic in a simply connected domain D except for a finite number of isolated singularities z₁, z₂, ..., zₙ inside a closed contour C in D, then ∫_C f(z) dz = 2πi ∑ (from k=1 to n) Res(f, z_k).
• The Residue Theorem is a powerful tool for evaluating contour integrals and has applications in various areas of mathematics, physics, and engineering.

Applications

• Complex analysis has applications in fluid dynamics, electromagnetism, quantum mechanics, and signal processing.
• Conformal Mapping: Analytic functions can be used to define conformal mappings, which preserve angles locally. Conformal mappings have applications in solving boundary value problems and in cartography.
• Evaluation of Real Integrals: The Residue Theorem can be used to evaluate certain types of real integrals that are difficult or impossible to evaluate using real calculus techniques.
• Special Functions: Complex analysis is used to study special functions such as the Gamma function, Bessel functions, and Legendre polynomials.