One solved region, carried everywhere
By the end of the last guide you owned a complete solution to one geometry: the Poisson integral formula takes any continuous boundary data on the unit circle and hands back the unique harmonic function inside the disk matching it. That solves the Dirichlet problem on the disk and nowhere else — yet. This guide is the lever that turns that single victory into a victory over almost every planar region, and the lever is conformal mapping. The whole rung has been quietly stockpiling the pieces; here they snap together.
The one fact that makes it all work is short enough to memorize: harmonicity survives a holomorphic change of variable. If u(w) is harmonic on some region and w = f(z) is a holomorphic map onto that region, then the composite u(f(z)) is harmonic back in the z-plane. The reason traces straight to the Cauchy-Riemann equations — a harmonic function is locally the real part of a holomorphic function, composing two holomorphic functions stays holomorphic, and the real part of a holomorphic function is again harmonic. So a conformal map does not merely move points; it moves entire solutions of Laplace's equation intact.
The Dirichlet problem on a general domain
Here is the recipe that cashes everything in. You are handed an awkward region D — a half-plane, a strip, the outside of an airfoil, a wedge — with temperatures or voltages prescribed along its boundary, and asked for the steady field inside. You cannot solve it directly, but you can transplant it. Find a conformal map f sending D onto the unit disk, push the boundary data along that map onto the circle, solve the now-standard disk problem with the Poisson integral, and pull the answer back. The conformality guarantees the pulled-back function is still harmonic and still matches the original boundary values.
- Map the region: find a conformal map w = f(z) carrying the domain D onto the unit disk (for a half-plane, the Cayley transform; for other shapes, a Mobius map, an exponential, or a Joukowski map).
- Carry the data: a boundary point z lands at the boundary point w = f(z), so the value prescribed at z becomes the value at f(z) on the circle.
- Solve on the disk: feed those circle values into the Poisson integral to get the unique harmonic U(w) inside the disk.
- Pull the answer back: the function u(z) = U(f(z)) is harmonic on D and solves the original problem, because harmonicity transfers along f.
And here is the line that makes the recipe astonishing rather than merely useful: the Riemann mapping theorem promises that step 1 always succeeds. Every simply connected region other than the whole plane can be conformally mapped onto the disk. So in principle the Dirichlet problem is solved on every reasonable simply connected planar region the moment it is solved on one disk — a sweeping result that the disk's modest Poisson formula could never have predicted on its own.
Harnack and the reflection principle
Two refinements sharpen the picture before we turn to physics. The first is Harnack's inequality, which controls how wildly a positive harmonic function can vary. On a disk it bounds the value at any interior point both above and below by the value at the center, with constants depending only on how close the point is to the boundary — never on the particular function. Concretely, a positive harmonic function cannot be huge at one interior point and tiny at a neighbor; positivity forces a gentle, quantitatively limited spread.
Harnack's payoff is a convergence theorem with no real-variable analogue this clean: if an increasing sequence of harmonic functions is bounded at even a single point, it converges everywhere on the region to a harmonic limit (or marches off to plus infinity uniformly). This is the engine behind Perron's method for building solutions on truly nasty domains, and it is the reason harmonic functions feel so much more rigid and well-behaved than arbitrary solutions of a PDE — they inherit the rigidity of their holomorphic partners.
The second refinement is the Schwarz reflection principle, and it is a small marvel of symmetry. Suppose a harmonic function is defined on the upper half of a disk and equals zero all along the real-axis diameter. Then it extends harmonically across that diameter into the lower half simply by reflecting it as an odd mirror image: u(x, -y) = -u(x, y). A boundary you thought was the edge of your domain turns out to be a hinge you can fold across, doubling the region for free. The same principle lets a holomorphic function that is real on a segment of the axis continue across it.
The complex potential: physics in one function
Now the physics, which is where all of this was secretly aimed. In a charge-free region the electrostatic potential is harmonic; in a source-free region the steady temperature is harmonic; in the steady, irrotational, incompressible flow of an ideal fluid the velocity potential is harmonic. Three different physical stories, one equation — Laplace's. And wherever a harmonic function u lives on a simply connected region, it has a harmonic conjugate v, and the two package into a single holomorphic function. That function is the complex potential, F(z) = u(x, y) + i v(x, y).
The genius of the complex potential is that its two halves carry the two halves of the physics. The real part u is the potential itself — its level curves u = constant are the equipotentials (lines of equal voltage, or isotherms of equal temperature). The imaginary part v is the conjugate, and its level curves v = constant are the streamlines — the field lines, the paths heat or charge or fluid actually travels along. Because u and v are harmonic conjugates, the Cauchy-Riemann equations force these two families to cross at right angles everywhere, exactly as physical field lines must cross equipotentials perpendicularly.
complex potential: F(z) = u(x,y) + i v(x,y), F holomorphic u = const -> equipotentials (equal voltage / isotherms) v = const -> streamlines (field lines / flow paths) C-R forces these two families to meet at right angles complex velocity / field: F'(z) = u_x - i u_y speed of flow = |F'(z)|; direction read off its argument
Even the derivative earns its keep. Differentiate F and you get F'(z) = u_x - i u_y, a single complex number whose modulus |F'(z)| is the strength of the field (the speed of the flow, the magnitude of the electric field) and whose argument tells you its direction. So one holomorphic function and its derivative encode the entire steady field: where it points, how strong it is, where its equipotentials and streamlines run. This is why complex analysis became the native language of two-dimensional potential theory — the algebra of holomorphic functions is the bookkeeping of the physics.
Flow past an obstacle, and the lift on a wing
Let the machine fly with a tiny worked picture. Uniform flow with speed U moving left-to-right has the simplest complex potential of all, F(z) = U z: its streamlines v = constant are the horizontal lines Im z = constant — a flat featureless current. Now drop a circular obstacle of radius a into the stream. Adding one cleverly chosen term gives F(z) = U (z + a^2 / z), and its streamlines automatically wrap around the circle |z| = a and rejoin smoothly downstream. We never solved a differential equation; we guessed a holomorphic function whose imaginary part draws the flow we wanted.
The cylinder is a warm-up; the real prize is the wing. The Joukowski map z = w + 1/w deforms a carefully placed circle into a smooth airfoil shape with a rounded nose and a sharp trailing edge. Because conformality carries the flow past the circle into a flow past the airfoil, you get the air flow over a wing almost for free — solve the easy circle, transplant. Layer in a circulation term and the streamlines speed up over the top of the wing and slow underneath, so by Bernoulli the pressure drops on top: the Kutta-Joukowski theorem reads the lift straight off the circulation in the complex potential.
The whole rung in one breath
Step back and the five guides of this rung tell one story. A harmonic function is the real part of a holomorphic function, so it inherits holomorphic rigidity: the mean-value property and the maximum principle from guide three, no local maxima inside, values determined entirely by the boundary. The Poisson integral from guide four turns that determination into an explicit formula on the disk. This guide ports that formula to any simply connected region by conformal mapping, sharpens it with Harnack and reflection, and reveals the physical engine underneath — the complex potential, where one holomorphic function carries an entire steady field.
Keep the honest boundaries in view so the craft stays trustworthy. The transplant needs simple connectivity — holes break it. The Riemann mapping theorem promises the map but, being non-constructive, does not write it down, so explicit maps remain the working currency. Conformal maps preserve angles and harmonicity but distort distances, and the entire two-into-one packaging of harmonic-plus-conjugate is a planar miracle with no higher-dimensional echo. None of these caveats weakens the method; they just mark exactly where it lives.
What you can now do is genuinely substantial. Hand yourself a hard planar region with prescribed boundary temperatures or voltages, and you can map it to a disk, solve with the Poisson integral, and pull the answer back with its field lines crossing its equipotentials at perfect right angles. Hand yourself an obstacle in a stream, and you can write a complex potential whose streamlines draw the flow around it. The harmonic world and the holomorphic world were never two worlds — and from this rung on, you can move freely between them.