Harmonic Functions

Complex Analysis · Axiom Academy

The real and imaginary parts of an analytic function — and why their level curves always cross at right angles. 1. Harmonic Means No Net Curvature A real function u(x,y) is harmonic when its Laplacian vanishes — the curvature it has in the x‑direction is exactly undone by its curvature in the y‑direction . the operator ^2 is the Laplacian Watch the saddle u=x^2-y^2 . Slice it along x and you get an upward parabola ( u_ xx =+2 ); slice along y and you get a downward one ( u_ yy =-2 ). The two bends are equal and opposite, so the readout ^2u sits at exactly 0 . 2. The Two Halves of an Analytic Function Where do harmonic functions come from? Every analytic function manufactures two of them. Take f(z)=z^2 and expand with z=x+iy : real part u + imaginary part v The animation peels f=z^2 into its real part u=x^2-y^2 and its imaginary part v=2xy — and both land in a box stamped ^2=0 . This is no coincidence; the Cauchy–Riemann equations guarantee it. because mixed partials agree ( v_ yx =v_ xy ), the two terms cancel. The same argument gives v_ xx +v_ yy =0 . When f=u+iv is analytic, v is called a harmonic conjugate of u . On a simply connected domain (one with no holes) every harmonic u has such a conjugate, unique up to an added constant. The same split for other analytic functions e^ z =e^ x y + i\,e^ x y : both e^ x y and e^ x y are harmonic. And z= 12 (x^2+y^2)+i z : the modulus part 12 (x^2+y^2) is harmonic away from the origin. 3. Level Curves Cross at Right Angles

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