--- title: "Harmonic Functions and What Analyticity Forces" description: "Show that the real and imaginary parts of an analytic function satisfy Laplace's equation, and construct harmonic conjugates with attention to the domain." canonical: https://lightmysky.com/learn/mathematics/harmonic-functions-and-what-analyticity-forces-mt_jqajP39XI4 source: https://lightmysky.com/learn/mathematics/harmonic-functions-and-what-analyticity-forces-mt_jqajP39XI4.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Harmonic Functions and What Analyticity Forces Show that the real and imaginary parts of an analytic function satisfy Laplace's equation, and construct harmonic conjugates with attention to the domain. Subject: Mathematics · Area: Complex Analysis · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/harmonic-functions-and-what-analyticity-forces-mt_jqajP39XI4 ## Ready when they can - Prove that analytic components are harmonic and state the maximum principle with its hypotheses - Construct a harmonic conjugate and state conditions for global existence - Explain the connection to steady heat and electrostatic potential, including conjugate level curves ## Lesson: Harmonic functions from analytic ones An open connected set D is called a domain. A real-valued function u with continuous second partial derivatives is harmonic on D if Δu = u_xx + u_yy = 0 throughout D. Compute Δu by differentiating twice in x, twice in y, then adding. At a point where u is harmonic, u_yy = -u_xx. **Example.** As in the Cauchy-Riemann lesson, write z = x + iy and use i² = -1. Expanding z² + 4z gives u = x² - y² + 4x and v = 2xy + 4y. The terms without i form u; the coefficient of i forms v. Here u_xx = 2 and u_yy = -2, while v_xx = v_yy = 0. Both Laplacians vanish. Here analytic means holomorphic on an open domain. Analytic functions have smooth components, so their mixed partials agree. From u_x = v_y and u_y = -v_x, obtain u_xx = v_yx and u_yy = -v_xy. Thus Δu = 0. Also v_xx = -u_yx and v_yy = u_xy, so Δv = 0. The maximum principle says that an interior local maximum or minimum forces a harmonic function to be constant on its connected domain. Constant harmonic functions can attain extrema inside. On a bounded domain, a harmonic function continuous on the closure attains its maximum and minimum on the boundary. Without those conditions, an extremum need not exist: u = x has no maximum on the whole plane. A harmonic conjugate v makes u + iv analytic. Solve v_y = u_x and v_x = -u_y. Integrate u_x with respect to y, keeping an unknown function h(x): v = ∫u_x dy + h(x). Use v_x = -u_y to find h'(x), then integrate it. Check both equations throughout the domain. On a connected domain, two conjugates of the same u differ by a real constant. **Example.** For u = x² - y² + 5x, integrating v_y = 2x + 5 gives v = (2x + 5)y + h(x). Since v_x must equal 2y, h'(x) = 0. Thus v = 2xy + 5y + C and f(z) = z² + 5z + iC. A specified value of v at one point fixes C. For u = ax + by, conjugate construction gives v = ay - bx + C and f(z) = (a - ib)z + iC. You can also recover the coefficient directly: if c = A + iB, then cz = Ax - By + i(Ay + Bx). Comparing coefficients of x and y in the real part determines A and B. Two harmonic functions need not form an analytic pair. For u = x and v = -y, both Laplacians vanish, but u_x = 1 differs from v_y = -1. For a proposed pair, calculate the residuals u_x - v_y and u_y + v_x. Both must vanish throughout the domain. With continuous first partials, these equations establish analyticity. For a real constant a, u = e^(ax) cos(ay) has u_xx = a²u and u_yy = -a²u, so Δu = 0. Differentiating cos(ay) in y gives -a sin(ay), then -a² cos(ay). A conjugate is v = e^(ax) sin(ay) + C: v_y = u_x and v_x = -u_y. Substitute a given value of v to find C before evaluating elsewhere. Angles are in radians. A harmonic u has a conjugate near each point. A global single-valued conjugate exists if every closed loop in the domain can shrink to a point inside it. A hole can obstruct this. For r = sqrt(x² + y²), u = log r is harmonic on the punctured plane. A local conjugate is the angle θ in radians. One circuit about 0 changes θ by 2π. Adding a constant cannot remove this change, so no global conjugate exists there. For heat diffusion with constant diffusivity κ > 0 and no internal sources, T_t = κΔT. Steady state means T_t = 0, so ΔT = 0. With a source term s, the model is T_t = κΔT + s. At steady state, ΔT = -s/κ. Thus steady temperature is not harmonic where s is nonzero. In a charge-free planar region with constant permittivity, electrostatic potential u satisfies Δu = 0. The gradient ∇u = (u_x,u_y) is normal to a regular level curve. If v is a conjugate, CR gives ∇u·∇v = 0. Where ∇u ≠ 0, the two level curves cross at right angles. The field E = -∇u is tangent to v's level curve and normal to u's. At a zero gradient, this regular-curve description need not apply. **Recap.** Analytic functions have harmonic components. Construct a conjugate with both Cauchy-Riemann equations, and check the domain before claiming a global one. *Original lesson, questions and flashcards written for this dataset, with revisions for mathematical accuracy.* ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss) ## Optional connections - [Laplace's Equation and the Maximum Principle](https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V) ## Opens up - [Contour Integrals Along Parametrised Paths](https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P)