From one demand, two equations fall out
The previous guide left us with a single, severe demand: for the complex derivative f'(z_0) to exist, the difference quotient (f(z) - f(z_0)) / (z - z_0) must approach the same limit no matter which direction z comes in from. That is directional independence, and it is far stronger than anything the real line ever asked of you. The Cauchy-Riemann equations are simply what happens when we cash out that demand in terms of u and v.
Write f = u + i v, where u(x, y) and v(x, y) are the real and imaginary parts. Approach z_0 along just two of the infinitely many directions. First slide in horizontally, letting z - z_0 = h be a small real number: the difference quotient becomes (change in f)/(change in x), and in the limit that is exactly u_x + i v_x. Now slide in vertically, letting z - z_0 = i k with k a small real number: dividing by i k contributes a factor 1/i = -i, and the limit becomes -i (u_y + i v_y) = v_y - i u_y.
If f is complex-differentiable at z_0, these two answers must be equal — same derivative, no matter the road. Matching real parts gives u_x = v_y; matching imaginary parts gives v_x = -u_y, that is u_y = -v_x. Those two identities are the Cauchy-Riemann equations. We have used only two directions, so they are merely a necessary condition — the necessary direction of the story. Whether they are also enough is a separate, more delicate question we settle below.
Side by side: the horizontal approach said f'(z_0) = u_x + i v_x, and the vertical approach said f'(z_0) = v_y - i u_y. Setting them equal is the whole derivation in one line — the real parts give u_x = v_y, the imaginary parts give -u_y = v_x. Notice the by-product the calculation hands you for free: whenever f' exists, you can read it straight off the horizontal limit as f'(z_0) = u_x + i v_x, a formula we lean on again and again.
Two worked checks: one passes, one fails
Equations are easier to trust once you have run them. Take f(z) = z^2. On the previous rung you split it into u = x^2 - y^2 and v = 2 x y. The partials are u_x = 2x, v_y = 2x, so u_x = v_y holds; and u_y = -2y, v_x = 2y, so u_y = -v_x holds too. Both equations check out everywhere, which fits what we already know: z^2 is differentiable on the whole plane, and indeed its derivative is u_x + i v_x = 2x + i (2y) = 2(x + i y) = 2z, exactly the answer the power rule predicts.
Now the cautionary tale: f(z) = z-bar, the conjugate, with u = x and v = -y. Here u_x = 1 but v_y = -1, so the very first equation u_x = v_y fails — and it fails at every point of the plane. Conjugation is smooth, gentle, and perfectly continuous as a map, yet it is complex-differentiable nowhere. This is the single most important example in the whole subject: it shows that real-smoothness is cheap and complex-differentiability is rare. The geometric reason is that z-bar reflects the plane, reversing orientation, and a complex derivative is never allowed to flip.
Necessary is not sufficient — the fine print
It is tempting to declare: "check the two equations, and you are done." Be honest — that is false. The Cauchy-Riemann equations only compared two directions; a sneaky function can satisfy them at a point and still have its difference quotient misbehave along a slanted or curved approach. The classic counterexample is the function equal to exp(-1/z^4) for z not 0 and to 0 at the origin: its partials satisfy the Cauchy-Riemann equations at 0, yet it is not differentiable there. So the equations alone are genuinely not enough.
The repair is small and reasonable. The sufficient condition says: if the four partials u_x, u_y, v_x, v_y exist and are continuous in a neighbourhood of z_0, and they satisfy the Cauchy-Riemann equations there, then f really is complex-differentiable at z_0 — with derivative f'(z_0) = u_x + i v_x. (An even leaner hypothesis works: it is enough that f be real-differentiable as a map of the plane and satisfy the equations. Continuity of the partials is the easy version to check.) In practice this is almost no extra burden, because the functions you meet — polynomials, exponentials, anything built from z by the usual operations — have continuous partials automatically.
What the equations are secretly saying: amplitwist
There is a gorgeous way to see why these two equations, and not some others, are the right ones. Any map of the plane has, at a point, a local linear approximation — a 2-by-2 real Jacobian matrix that tells you how tiny vectors get stretched and turned. For a general smooth map that matrix can do anything: stretch the x-direction differently from the y-direction, shear, reflect. The Cauchy-Riemann equations are exactly the conditions that force this matrix to be of one very special shape.
Jacobian of f = u + i v :
[ u_x u_y ]
[ v_x v_y ]
impose u_x = v_y = a and u_y = -v_x = -b :
[ a -b ]
[ b a ] = a rotation-and-scaling, by angle arg(a + i b) and factor sqrt(a^2 + b^2)A matrix of the form [a, -b; b, a] is precisely what you get from multiplying complex numbers by a + i b: it rotates by the argument of a + i b and scales by its modulus, with no shearing and no flipping. This is the amplitwist — Needham's vivid name for "amplify and twist." So the real content of the Cauchy-Riemann equations is: to first order, a complex-differentiable map acts on every tiny neighbourhood as a single rotation-and-scaling, the same in all directions. The complex number doing that job is the derivative f'(z_0), with a + i b = u_x + i v_x. The next guide builds this amplitwist picture out in full; here it explains why the equations look the way they do.
A bonus the equations hand you: harmonic parts
Once you trust that a holomorphic function — one that is complex-differentiable on an open set — is automatically as smooth as you like (a fact the later rung proves, and which has no real-variable analogue), the Cauchy-Riemann equations let you squeeze out a surprising consequence about u and v on their own. Differentiate u_x = v_y with respect to x, and u_y = -v_x with respect to y, then add. The mixed partials of v cancel because order does not matter, and you are left with u_xx + u_yy = 0. Run the same trick the other way and v satisfies it too.
The equation u_xx + u_yy = 0 is Laplace's equation, and any function obeying it is called harmonic. So both the real and imaginary parts of a holomorphic function are harmonic — they cannot wobble freely; they are governed by the smoothest, most rigid equation in physics, the one that describes steady temperature, electrostatic potential, and ideal fluid flow. This is why complex analysis is a workshop for two-dimensional physics. The two parts are not merely each harmonic but are bound to each other: v is a harmonic conjugate of u, meaning the pair u, v solves the Cauchy-Riemann equations together.
That binding is constructive: given a harmonic u, you can reconstruct a partner v (unique up to an added constant) and hence recover the whole holomorphic f, almost from its real part alone. The mechanics — integrate v_y = u_x, then fix the leftover function of x using v_x = -u_y — are the subject of the rung's final guide. For now, marvel at the chain you have forged: a single complex derivative forces the Cauchy-Riemann equations, which force each part to be harmonic, which ties the two parts into one inseparable holomorphic whole.