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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Analytic functions carry two harmonic halves, glued by Cauchy-Riemann and finished with a conjugate.
2 · Watch
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.