Table of Contents
- Introduction
- What Are Complex Numbers?
- Algebra of Complex Numbers
- Complex Plane and Polar Representation
- Euler’s Formula and Roots of Unity
- Complex Functions: Definition and Examples
- Analytic Functions and the Cauchy-Riemann Conditions
- Complex Differentiation and Holomorphicity
- Complex Integration and Cauchy’s Theorem
- Laurent Series and Residue Calculus
- Singularities and Poles
- Branch Cuts and Multivalued Functions
- Conformal Mapping
- Applications in Physics
- Conclusion
1. Introduction
Complex numbers and functions play a foundational role in physics, mathematics, and engineering. From oscillations and wave mechanics to quantum theory and electrical engineering, they offer a powerful framework for describing both algebraic and geometric behavior.
2. What Are Complex Numbers?
A complex number is defined as:
\[
z = x + iy
\]
Where:
- \( x \): real part \( \text{Re}(z) \)
- \( y \): imaginary part \( \text{Im}(z) \)
- \( i \): imaginary unit with \( i^2 = -1 \)
3. Algebra of Complex Numbers
- Addition: \( (a + ib) + (c + id) = (a + c) + i(b + d) \)
- Multiplication:
\[
(a + ib)(c + id) = (ac – bd) + i(ad + bc)
\] - Complex conjugate:
\[
\bar{z} = x – iy
\] - Modulus:
\[
|z| = \sqrt{x^2 + y^2}
\]
4. Complex Plane and Polar Representation
A complex number can be represented as a point in the complex plane.
In polar coordinates:
\[
z = r(\cos \theta + i \sin \theta) = re^{i\theta}
\]
Where:
- \( r = |z| \): modulus
- \( \theta = \arg(z) \): argument (angle with positive real axis)
5. Euler’s Formula and Roots of Unity
Euler’s formula:
\[
e^{i\theta} = \cos \theta + i \sin \theta
\]
Roots of unity are solutions to \( z^n = 1 \), given by:
\[
z_k = e^{2\pi i k/n}, \quad k = 0, 1, \dots, n-1
\]
They form vertices of a regular \( n \)-gon on the unit circle.
6. Complex Functions: Definition and Examples
A complex function maps complex inputs to complex outputs:
\[
f: \mathbb{C} \to \mathbb{C}
\]
Examples:
- \( f(z) = z^2 \)
- \( f(z) = e^z \)
- \( f(z) = \frac{1}{z} \)
7. Analytic Functions and the Cauchy-Riemann Conditions
A function \( f(z) = u(x, y) + i v(x, y) \) is analytic if it is differentiable in a neighborhood.
Cauchy-Riemann equations:
\[
\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}
\]
If satisfied, the function is holomorphic and locally power-expandable.
8. Complex Differentiation and Holomorphicity
A function is holomorphic at a point if the complex derivative exists:
\[
f'(z) = \lim_{\Delta z \to 0} \frac{f(z + \Delta z) – f(z)}{\Delta z}
\]
Unlike real analysis, differentiability implies infinite differentiability and analyticity.
9. Complex Integration and Cauchy’s Theorem
Integral over a path \( C \):
\[
\int_C f(z) \, dz
\]
If \( f \) is analytic in a region bounded by \( C \), then:
\[
\oint_C f(z) \, dz = 0
\]
(Cauchy’s theorem)
10. Laurent Series and Residue Calculus
If \( f(z) \) has a singularity at \( z_0 \), we can expand it as a Laurent series:
\[
f(z) = \sum_{n=-\infty}^{\infty} a_n (z – z_0)^n
\]
The residue at \( z_0 \): \( \text{Res}{z=z_0} f = a{-1} \)
11. Singularities and Poles
- Removable: singularity where function is bounded
- Pole: singularity where \( f(z) \to \infty \) like \( 1/(z – z_0)^n \)
- Essential: behavior cannot be captured by finite pole expansion
12. Branch Cuts and Multivalued Functions
Functions like \( \log z \), \( z^{1/n} \), etc., are multivalued. We define branch cuts to make them single-valued in specified regions.
Example: branch cut for \( \log z \) usually taken along the negative real axis.
13. Conformal Mapping
A function is conformal if it preserves angles locally. If \( f \) is holomorphic and \( f'(z) \neq 0 \), it is conformal at \( z \).
Used in:
- Electrostatics
- Fluid dynamics
- String theory
14. Applications in Physics
- Quantum mechanics: complex wavefunctions
- Electromagnetism: phasors and impedance
- Relativity: complex representation of spacetime rotations
- Signal processing: Fourier transforms
- String theory: complex manifolds and conformal fields
15. Conclusion
Complex numbers and functions extend the real number system into a richer and more flexible structure. They provide elegant tools for solving problems in physics, engineering, and pure mathematics.
From Euler’s identity to complex contour integrals, the complex domain reveals deeper structures and powerful methods that underlie many natural phenomena.
