Harmonic Functions, Harmonic Conjugates & Milne-Thomson Construction Method
Idea in plain words
Harmonic Function: A real function φ(x, y) satisfying Laplace's 2D Partial Differential Equation: ∇²φ = ∂²φ/∂x² + ∂²φ/∂y² = 0.
Exam tip
High-yield exam topic (Very High).
Explanation
Laplace's equation models steady-state temperature distributions, electrostatic potential, and irrotational inviscid fluid flows. In two dimensions, every harmonic function corresponds to the real part of an analytic complex function. The Milne-Thomson method leverages the identity f'(z) = u_x - i u_y on the real axis (y = 0), reducing multi-variable partial integration to single-variable complex antiderivatives.
Key points
- Theorem: If f(z) = u + iv is analytic, then BOTH u and v are harmonic functions. v is the harmonic conjugate of u.
- Orthogonal System: The families of curves u(x, y) = c₁ and v(x, y) = c₂ intersect orthogonally at all points where f'(z) ≠ 0: (m₁ · m₂ = -1).
- Finding Harmonic Conjugate v(x, y): dv = (∂v/∂x) dx + (∂v/∂y) dy = (-∂u/∂y) dx + (∂u/∂x) dy (exact differential, integrated directly).
- Milne-Thomson Method: Directly constructs f(z) without finding v:
-
- Compute u_x = ∂u/∂x = φ₁(x, y) and u_y = ∂u/∂y = φ₂(x, y).
-
- Substitute x = z and y = 0.
-
- Integrate: f(z) = ∫ [φ₁(z, 0) - i φ₂(z, 0)] dz + C.
- When v(x, y) is given: f(z) = ∫ [ψ₁(z, 0) + i ψ₂(z, 0)] dz + C where ψ₁ = ∂v/∂y and ψ₂ = ∂v/∂x.
Example
Example
Prove that u = 2x - x³ + 3xy² is harmonic: u_x = 2 - 3x² + 3y², u_{xx} = -6x. u_y = 6xy, u_{yy} = 6x. ∇²u = u_{xx} + u_{yy} = -6x + 6x = 0. By Milne-Thomson: f'(z) = (2 - 3z²) - i(0) => f(z) = 2z - z³ + C.
Code examples
// Milne-Thomson Method Example:
// Given: u(x, y) = e^x (x cos y - y sin y)
// Step 1: Find partial derivatives:
// u_x = e^x (x cos y - y sin y) + e^x (cos y)
// u_y = e^x (-x sin y - sin y - y cos y)
// Step 2: Put x = z and y = 0:
// φ1(z, 0) = u_x(z, 0) = e^z (z * 1 - 0) + e^z (1) = z e^z + e^z
// φ2(z, 0) = u_y(z, 0) = e^z (0 - 0 - 0) = 0
// Step 3: By Milne-Thomson Formula:
// f'(z) = φ1(z, 0) - i φ2(z, 0) = z e^z + e^z
// Step 4: Integrate with respect to z:
// f(z) = ∫ (z e^z + e^z) dz + C
// = [ z e^z - e^z + e^z ] + C = z e^z + C
// Verify:
// z e^z = (x + iy) e^{x + iy} = (x + iy) e^x (cos y + i sin y)
// = e^x [ (x cos y - y sin y) + i (y cos y + x sin y) ]
// Real part is exactly u(x, y)! (Harmonic conjugate v = e^x (y cos y + x sin y))
Exam questions
- [8 Marks] Show that u = x³ - 3xy² + 3x² - 3y² + 1 is harmonic. Find its harmonic conjugate v and express f(z) in terms of z.
- [8 Marks] If u = e^{2x} (x cos 2y - y sin 2y), find the analytic function f(z) using the Milne-Thomson method.
- [7 Marks] Prove that the families of curves u(x, y) = c₁ and v(x, y) = c₂ cut orthogonally if f(z) = u + iv is an analytic function.