Cauchy-Riemann equations

In complex analysis, the Cauchy-Riemann equations are one of the of the basic objects of the theory. These equations are a set of $\scriptstyle 2n$ partial differential equations, where $\scriptstyle n$ is a positive integer, expressing a necessary and sufficient condition between the real and imaginary part of a complex valued function of $\scriptstyle 2n$ variables for the given function to be a holomorphic one.