Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2013 / Analysis / 4 / a

Under the assumed continuous real differentiability, the necessary and sufficient conditions are the Cauchy-Riemann equations at every point of $U$:Necessity follows by computing the complex derivative along real and imaginary increments. For sufficiency, real differentiability gives the first-order expansionThe Cauchy-Riemann equations turn the linear part into $(u_x+i{v}_x)h$, so the complex derivative exists and equals $u_x+i{v}_x$. Thus $f$ is a holomorphic function on $U$.