The complex analytic functions, those that admit power series expansions in one complex variable, can also be characterized as solution of a certain elliptic partial differential equation, namely the Cauchy-Riemann equation(s).

Real analytic functions lack such characterizations. Also, the celebrated Cauchy residue formula, is a manifestation of the fact that the Cauchy kernel

$$\frac{1}{\pi\boldsymbol{i} z} $$

is a fundamental solution of the Cauchy-Riemann operator.

The complex analytic functions, those that admit power series expansions in one complex variable, can also be characterized as solutions of a certain elliptic partial differential equation, namely the Cauchy-Riemann equation(s).

Real analytic functions lack such characterizations. Also, the celebrated Cauchy residue formula is a manifestation of the fact that the Cauchy kernel

$$\frac{1}{\pi\boldsymbol{i} z} $$

is a fundamental solution of the Cauchy-Riemann operator.