Function of Complex Variables

Unit IV: Function of Complex Variables

4.1 Complex Numbers Review

A complex number z has the form z = x + iy, where x is the real part (Re(z)), y is the imaginary part (Im(z)), and i = √(-1). Complex numbers can also be expressed in polar form: z = r(cosθ + i sinθ) = re^iθ, where r = |z| is the modulus and θ is the argument.

4.2 Functions of a Complex Variable

A function f(z) maps complex numbers to complex numbers. We write f(z) = u(x,y) + iv(x,y), where u and v are real-valued functions of two real variables x and y.

Example: f(z) = z²

f(z) = (x + iy)² = x² - y² + 2ixy
So u(x,y) = x² - y² and v(x,y) = 2xy

4.3 Analytic Functions

A function f(z) is analytic (holomorphic) at a point z₀ if it is differentiable at z₀ and at every point in some neighborhood of z₀. The derivative is defined as:

f'(z) = lim_Δz→0 [f(z + Δz) - f(z)] / Δz

For this limit to exist, it must be the same regardless of the direction from which Δz approaches zero in the complex plane.

4.4 Cauchy-Riemann Equations

The necessary and sufficient conditions for f(z) = u + iv to be analytic are the Cauchy-Riemann equations:

∂u/∂x = ∂v/∂y   and   ∂u/∂y = -∂v/∂x

Example: Verify f(z) = z² is analytic

u = x² - y², v = 2xy
∂u/∂x = 2x, ∂v/∂y = 2x ⇒ ∂u/∂x = ∂v/∂y ✓
∂u/∂y = -2y, ∂v/∂x = 2y ⇒ ∂u/∂y = -∂v/∂x ✓
Both conditions satisfied, so f(z) = z² is analytic everywhere.

4.5 Harmonic Functions

A function φ(x,y) is harmonic if it satisfies Laplace's equation:

∂²φ/∂x² + ∂²φ/∂y² = 0

If f(z) = u + iv is analytic, then both u and v are harmonic functions. They are called harmonic conjugates of each other.

Example: Show u = x² - y² is harmonic and find its conjugate

∂²u/∂x² = 2, ∂²u/∂y² = -2
Sum = 2 + (-2) = 0 ✓ (harmonic)
Using C-R equations: ∂v/∂y = ∂u/∂x = 2x, so v = 2xy + g(x)
∂v/∂x = 2y + g'(x) = -∂u/∂y = 2y, so g'(x) = 0, g(x) = C
Conjugate: v = 2xy + C

4.6 Elementary Complex Functions

Key complex functions include:

• Exponential: e^z = e^x(cos y + i sin y)
• Trigonometric: sin z = (e^iz - e^-iz)/(2i), cos z = (e^iz + e^-iz)/2
• Logarithmic: ln z = ln|z| + i(arg z + 2nπ) (multivalued)

4.7 Conformal Mapping

A conformal mapping preserves angles between curves at every point where the derivative is nonzero. If f(z) is analytic and f'(z) ≠ 0, then f is conformal at z.

Common Conformal Mappings:

The Möbius (bilinear) transformation w = (az + b)/(cz + d) where ad - bc ≠ 0 maps circles and lines to circles and lines, and is widely used in complex analysis and applications.