Splitting f and meeting Laplace's equation
You have spent several rungs treating a holomorphic function f as a single complex object, but from the very start you also knew how to take it apart: write f(z) = u(x, y) + i v(x, y), splitting the one complex value into a real part u and an imaginary part v, each an ordinary real-valued function of the two real coordinates x and y. This guide asks a simple-sounding question with a surprisingly deep answer — what equation does u, all by itself, satisfy? The pieces are already in your hands; we just have to combine them.
The combining tool is the pair of equations that defined holomorphy in the first place, the Cauchy-Riemann equations u_x = v_y and u_y = -v_x. The trick is to differentiate them a second time and then add. Differentiate the first equation with respect to x to get u_xx = v_yx, and differentiate the second with respect to y to get u_yy = -v_xy. The right-hand sides are the mixed second partials of v, v_yx and v_xy, and as long as v is smooth enough that the order of differentiation does not matter, those two mixed partials are equal. So when you add the two lines, the v terms cancel exactly.
u_x = v_y --d/dx--> u_xx = v_yx
u_y = -v_x --d/dy--> u_yy = -v_xy
add: u_xx + u_yy = v_yx - v_xy = 0 (mixed partials agree)
so u_xx + u_yy = 0 <-- Laplace's equation
likewise v_xx + v_yy = 0 (same trick on v)What survives is u_xx + u_yy = 0, the second-order partial differential equation called Laplace's equation. A real-valued function that satisfies it on an open region is called harmonic. Running the very same second-derivative-and-add argument on the imaginary part gives v_xx + v_yy = 0 as well. So the headline, proved in three lines, is this: the real and imaginary parts of any holomorphic function are both harmonic.
What harmonic means, in a picture
Laplace's equation looks abstract, so it helps to read u_xx + u_yy = 0 physically before going further. The second derivative u_xx measures how u curves as you walk in the x-direction: positive means u is bending upward like a valley, negative means it bows over like a hill. Laplace's equation says u_xx and u_yy must be exactly negatives of each other at every point. In plain words: however much the surface curves up along one axis, it must curve down by precisely the same amount along the perpendicular axis. A harmonic graph can never be a pure bowl or a pure dome — the two curvatures are forced into perfect balance.
This balance has a famous everyday face: a harmonic function is the steady-state temperature of a thin plate. Heat a metal sheet however you like along its edge, wait until nothing changes anymore, and the temperature u(x, y) you measure inside is harmonic. The physical reason matches the curvature picture exactly — in steady state no point can be a strict hot spot or cold spot, because heat would still be flowing into or out of it. Every interior temperature is the perfectly settled average of the temperatures around it. That averaging idea is the soul of harmonicity, and the third guide of this rung will make it precise as the mean-value property.
Harmonic functions inherit holomorphy's superpowers
Here is where being the part of a holomorphic function pays enormous dividends. Recall the single most astonishing fact from the Cauchy integral rung: one complex derivative forces infinite differentiability, so a holomorphic function automatically has derivatives of every order — there is no real-variable analogue of that rigidity at all. Because a harmonic u (on a nice region) is the real part of some holomorphic f, u silently inherits this. A function that you only assumed had two derivatives — just enough to write u_xx + u_yy — turns out, for free, to be infinitely smooth.
The same inheritance hands you rigidity. A holomorphic function is pinned down by astonishingly little local data — its values on a tiny disk, even on a convergent sequence of points, determine it everywhere on a connected region. Harmonic functions are just as stiff. You cannot wiggle a harmonic function in a small patch while leaving it alone outside; doing so would mean assembling a new holomorphic function that agreed with the old one on a big set, and the identity theorem forbids that. A harmonic temperature distribution is not a floppy thing you can dent locally — it is a single rigid object held in place by its values on the boundary.
Most consequentially, harmonic functions inherit the maximum principle. For holomorphic f the maximum modulus principle said |f| can attain its peak only on the boundary, never strictly inside, unless f is constant. The harmonic version is even more intuitive: a non-constant harmonic u attains both its maximum AND its minimum value only on the boundary of its region. The steady-state temperature picture makes this almost obvious — the hottest and coldest points of a settled plate sit on its edge, because any strict interior extreme would still be exchanging heat with its neighbours. This is the maximum principle for harmonic functions, and it is the workhorse behind every uniqueness theorem in this rung.
Two harmonic functions, locked together
Now look again at the split f = u + i v. We have shown u and v are each harmonic, but they are not two strangers who happen both to be harmonic — they are bound to each other by the Cauchy-Riemann equations u_x = v_y, u_y = -v_x. When u and v are tied this way, v is called a harmonic conjugate of u. The name signals a precise, asymmetric relationship: given u, its conjugate v is determined up to an additive constant, but you are not free to pick v independently — the two are gears that must mesh.
A quick concrete case makes the locking tangible. Take u(x, y) = x^2 - y^2; check it is harmonic: u_xx = 2 and u_yy = -2, which sum to 0, good. What v must accompany it? Cauchy-Riemann demands v_y = u_x = 2x and v_x = -u_y = 2y. Integrate the first in y to get v = 2xy plus a function of x alone; differentiating that in x and matching the second equation forces the extra function to be a constant. So v = 2xy, and indeed u + i v = (x^2 - y^2) + i(2xy) = z^2 — the conjugate of the real part of z^2 is exactly its imaginary part, just as it had to be. The full recipe for this hunt, including the holes that can trip it, is the next guide.
Why this matters: physics speaks harmonic
The reason this whole rung exists is that an enormous slice of two-dimensional physics is written in Laplace's equation. Steady heat flow, the electrostatic potential in a charge-free region, the velocity potential of an ideal incompressible irrotational fluid, the displacement of a stretched membrane — all of these are harmonic functions. The pairing u + i v packages a physical problem and its 'shadow' into a single holomorphic object called the complex potential, and the fifth guide of this rung will cash that out in full.
Here is the elegance of the pairing in a physical reading. If u is the electrostatic potential, its level curves u = constant are the equipotential lines, the curves along which a free charge feels no push. The conjugate v turns out to have level curves v = constant that are precisely the field lines — the paths a charge actually travels. And because u + i v is holomorphic, the Cauchy-Riemann relations force these two families to cross at right angles everywhere (away from critical points). The orthogonality of field lines and equipotentials, a fact a physics course states as given, is here a free consequence of u and v being conjugate.
All of this sets up the central engineering question of the rung, the Dirichlet problem: given the values a harmonic function must take on the boundary of a region — the temperatures clamped on a plate's edge, the voltages on a conductor — find the unique harmonic function inside that matches them. The maximum principle from earlier already guarantees that answer is unique. What remains is to actually build it, and the Poisson integral in the fourth guide will write it down explicitly on a disk. The thread of this rung is now visible end to end: split, conjugate, the maximum principle, then solve.