Chapter 2: Analytic Functions and Complex Calculus | Complex Functions and Special Functions

2.1 Complex Differentiation and Cauchy-Riemann Equations

A complex function $f(z)$ is differentiable at point $z_0$ if the limit $\lim_{h \to 0} \frac{f(z_0+h) - f(z_0)}{h}$ exists independently of how $h$ approaches zero.

📐 Theorem: Cauchy-Riemann Equations
Necessary and sufficient condition for $f(z) = u(x,y) + iv(x,y)$ to be analytic: $$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$ Calculation of complex derivative: $$f'(z) = \frac{\partial u}{\partial x} + i\frac{\partial v}{\partial x} = \frac{\partial v}{\partial y} - i\frac{\partial u}{\partial y}$$

💻 Code Example 1: Verification of Cauchy-Riemann Equations

Python Implementation: Verification of Cauchy-Riemann Equations

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt
from scipy.misc import derivative

def f_analytic(z):
    """Analytic function: f(z) = z^2"""
    return z**2

def f_not_analytic(z):
    """Non-analytic function: f(z) = z̄ (complex conjugate)"""
    return np.conj(z)

# Separate real and imaginary parts
def extract_uv(f, x, y):
    z = x + 1j*y
    w = f(z)
    return w.real, w.imag

# Numerical calculation of partial derivatives
def check_cauchy_riemann(f, x0, y0, h=1e-5):
    u, v = extract_uv(f, x0, y0)

    # ∂u/∂x
    u_xp, _ = extract_uv(f, x0+h, y0)
    u_xm, _ = extract_uv(f, x0-h, y0)
    du_dx = (u_xp - u_xm) / (2*h)

    # ∂u/∂y
    u_yp, _ = extract_uv(f, x0, y0+h)
    u_ym, _ = extract_uv(f, x0, y0-h)
    du_dy = (u_yp - u_ym) / (2*h)

    # ∂v/∂x
    _, v_xp = extract_uv(f, x0+h, y0)
    _, v_xm = extract_uv(f, x0-h, y0)
    dv_dx = (v_xp - v_xm) / (2*h)

    # ∂v/∂y
    _, v_yp = extract_uv(f, x0, y0+h)
    _, v_ym = extract_uv(f, x0, y0-h)
    dv_dy = (v_yp - v_ym) / (2*h)

    return du_dx, du_dy, dv_dx, dv_dy

# Test point
x0, y0 = 1.5, 2.0

print("=== Analytic function: f(z) = z^2 ===")
du_dx, du_dy, dv_dx, dv_dy = check_cauchy_riemann(f_analytic, x0, y0)
print(f"∂u/∂x = {du_dx:.6f}")
print(f"∂v/∂y = {dv_dy:.6f}")
print(f"∂u/∂x - ∂v/∂y = {du_dx - dv_dy:.6e} (should be ~0)")
print(f"\n∂u/∂y = {du_dy:.6f}")
print(f"-∂v/∂x = {-dv_dx:.6f}")
print(f"∂u/∂y - (-∂v/∂x) = {du_dy - (-dv_dx):.6e} (should be ~0)")

print("\n\n=== Non-analytic function: f(z) = z̄ ===")
du_dx, du_dy, dv_dx, dv_dy = check_cauchy_riemann(f_not_analytic, x0, y0)
print(f"∂u/∂x = {du_dx:.6f}")
print(f"∂v/∂y = {dv_dy:.6f}")
print(f"∂u/∂x - ∂v/∂y = {du_dx - dv_dy:.6f} (NOT ~0)")
print(f"\n∂u/∂y = {du_dy:.6f}")
print(f"-∂v/∂x = {-dv_dx:.6f}")
print(f"∂u/∂y - (-∂v/∂x) = {du_dy - (-dv_dx):.6f} (NOT ~0)")

# Visualization omitted (see original code)

📌 Note: For analytic functions, the contour lines of real and imaginary parts are orthogonal (geometric meaning of Cauchy-Riemann equations).

2.2 Examples and Properties of Analytic Functions

Many complex functions are analytic, but functions involving complex conjugate or absolute value are not analytic.

📐 Theorem: Analytic Functions
Examples of analytic functions:

Polynomials: $z^n$, $a_n z^n + \cdots + a_1 z + a_0$
Exponential function: $e^z$
Trigonometric functions: $\sin z$, $\cos z$
Rational functions: $\frac{P(z)}{Q(z)}$ (in region where $Q(z) \neq 0$)

Examples of non-analytic functions:

Complex conjugate: $\bar{z}$
Real/Imaginary part: $\mathrm{Re}(z)$, $\mathrm{Im}(z)$
Absolute value: $|z|$

2.3 Calculation of Complex Integrals

Complex integrals are defined as path integrals: $\int_C f(z) dz = \int_a^b f(z(t)) z'(t) dt$

📐 Theorem: Complex Line Integral
$$\int_C f(z) dz = \int_a^b f(z(t)) \frac{dz}{dt} dt$$ where $z(t)$ is parametric representation of path $C$ $(a \leq t \leq b)$

2.4 Cauchy's Integral Theorem

The integral of an analytic function along a closed curve is zero. This is known as Cauchy's integral theorem.

📐 Theorem: Cauchy's Integral Theorem
$$\oint_C f(z) dz = 0$$ where $f(z)$ is analytic inside closed curve $C$

2.5 Cauchy's Integral Formula

The value of an analytic function can be obtained from values on the surrounding closed curve.

📐 Theorem: Cauchy's Integral Formula
Cauchy's integral formula: $$f(z_0) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z - z_0} dz$$ Formula for derivatives: $$f^{(n)}(z_0) = \frac{n!}{2\pi i} \oint_C \frac{f(z)}{(z - z_0)^{n+1}} dz$$

2.6 Conformal Mapping and Harmonic Functions

Analytic functions are angle-preserving mappings (conformal mappings). Also, the real and imaginary parts of analytic functions are harmonic functions (solutions to Laplace's equation).

📐 Theorem: Conformal Mapping and Harmonic Functions
Conformal mapping: Analytic function $w = f(z)$ preserves angles
Harmonic functions: If $f(z) = u + iv$ is analytic, then $$\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$ $$\nabla^2 v = \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} = 0$$

📝 Chapter Exercises

✏️ Exercises

For $f(z) = z^3$, verify the Cauchy-Riemann equations at point $z_0 = 1+i$.
Calculate $\oint_C z^n dz$ where $C$ is unit circle centered at origin and $n$ is an integer.
For $f(z) = \frac{1}{z-2}$, calculate integral $\oint_C f(z) dz$ along unit circle and discuss relation to Cauchy's integral theorem.
Find what curve the line $x=1$ is mapped to by conformal mapping $w = z^2$.

🔗 References

Ahlfors, L. V. (1979). Complex Analysis. McGraw-Hill.
Stein, E. M., & Shakarchi, R. (2003). Complex Analysis. Princeton University Press.

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