--- title: "Harmonic Functions and What Analyticity Forces" description: "Show the real and imaginary parts of an analytic function each satisfy Laplace's equation, and recover one from the other." 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: 2026-09-12 --- > **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 the real and imaginary parts of an analytic function each satisfy Laplace's equation, and recover one from the other. 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 the real part of an analytic function is harmonic - Construct a harmonic conjugate for a given harmonic function - Explain why this links complex analysis to steady-state heat and potential problems ## Lesson: Harmonic functions from analytic ones A function u is harmonic when its Laplacian vanishes: u_xx plus u_yy equals 0 everywhere in the domain. Such functions have no peaks inside; maxima sit on the boundary. They describe steady heat, electrostatic potential, and ideal fluid flow. **Example.** Take f of z equal to z squared, split as (x plus iy) squared. The real part is x squared minus y squared and the imaginary part is 2xy. Check: 2 plus minus 2 is 0, so both parts are harmonic. This always works: the real part of any analytic function is harmonic. Differentiate u_x equal to v_y in x and u_y equal to minus v_x in y, and the mixed partials cancel. The Cauchy-Riemann equations force the Laplacian to zero. **Tip.** The partner v is called the harmonic conjugate. Find it from v_y equal to u_x and v_x equal to minus u_y, up to an added constant. For u equal 2xy that gives v equal y squared minus x squared, and for x cubed minus 3x times y squared it gives 3x squared times y minus y cubed. **Recap.** Analytic functions carry two harmonic halves, glued by Cauchy-Riemann and finished with a conjugate. ## Practice 17 questions on this page, each with its working shown. ## Needs first - [Laplace's Equation and the Maximum Principle](https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V) - [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss) ## Opens up - [Contour Integrals Along Parametrised Paths](https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P)