Complex Analysis

Cauchy–Riemann equations

The Cauchy–Riemann equations are the necessary conditions that a complex function must satisfy in order to be complex-differentiable. Write f(z)=u(x,y)+iv(x,y) with z=x+iy. If the limit f'(z_0)=\lim_{h\to 0}\frac{f(z_0+h)-f(z_0)}{h} exists, its value must be the same whether h tends to zero along the real axis or along the imaginary axis. Comparing these two directional derivatives gives u_x=v_y and u_y=-v_x, and at such a point f'(z_0)=u_x+iv_x=v_y-iu_y. The equations are sufficient as well when u and v have continuous first partial derivatives on an open set. To test a given function, substitute its partial derivatives into both equations; if they hold identically, the function is analytic, and f' can then be evaluated at any point by substitution, with |f'|^2=(\text{Re}\,f')^2+(\text{Im}\,f')^2 used to find the modulus.

Analytic (holomorphic) at a point means differentiable on an open disc around that point, so analyticity is a property of a neighbourhood and not of a single point. A function may satisfy the Cauchy–Riemann equations at isolated points and still fail to be analytic there. For example, f=x^2+iy^2 satisfies the equations only on the line y=x, so it is differentiable on that line but nowhere analytic. If the equations fail at points arbitrarily close to z_0, then f cannot be differentiable on a neighbourhood of z_0, and hence it is not analytic at z_0.

Complex differentiability at an isolated point is possible, and it can be detected with the Wirtinger derivative: f is differentiable at z only if \frac{\partial f}{\partial \bar z}=\frac12(\partial_x+i\partial_y)f=0 there. For f(z)=|z|^2=z\bar z, this derivative equals z, which vanishes only at the origin. Since |h|^2/h=\bar h\to 0 as h\to 0, f'(0)=0 exists, while f is differentiable nowhere else. Entire functions, which are analytic on all of \mathbb{C}, include polynomials, \sin z and e^{-z^2}; by contrast, e^{1/z} has an essential singularity at 0 and is not differentiable there.