Cauchy-Riemann Equations

The partial-derivative test for when a complex function is holomorphic.

The idea

Cauchy-Riemann equations. Write a complex function as $f(z) = u(x, y) + i\,v(x, y)$, with $z = x + iy$, so that $u$ and $v$ are real functions of the two real variables $x$ and $y$. If $f$ is complex-differentiable at a point, then at that point $u_{x} = v_{y}, \qquad u_{y} = -v_{x}.$ Conversely, if the four partials exist, are continuous on an open set, and satisfy both equations throughout it, then $f$ is complex-differentiable at every point of that set, which is what holomorphic on the set means.

The equations come from the definition of the derivative. $f'(z)$ is a single number that the difference quotient must approach however the increment $h$ shrinks to $0$. Take $h$ real, so the point moves horizontally, and the quotient tends to $u_{x} + i\,v_{x}$. Take $h$ purely imaginary, so the point moves vertically, and dividing by the imaginary increment turns the quotient into $v_{y} - i\,u_{y}$. These are two expressions for one complex number, so their real parts agree and their imaginary parts agree, which is exactly the pair of equations above.

The equations are therefore not an extra condition imposed on top of differentiability: they are what complex differentiability requires of $f$ read as a map of the plane.

Ways to work on it

• Walkthrough. Derive and check the equations u_x = v_y, u_y = -v_x on a simple power.
• Practice. Evaluate a Cauchy-Riemann partial at a point.
• Hardest. Show that a familiar function fails the equations at every point.

Not sure where to start? Take the ten-question placement test.