Components of a 2D vector field satisfy Cauchy Riemann conditions implies the finite transformation is holomorphic

Question

I am reading M. Schottenloher's book on Conformal field theory.

https://www.mathematik.uni-muenchen.de/~schotten/LNP-cft-pdf/01_978-3-540-68625-5_Ch01_23-08-08.pdf

On page 19 after Proposition 1.12, he considers a vector field $X=(u,v)$, where $u_x=v_y$, $u_y=-v_x$. And then he claims that the one-parameter groups for $X$ are also holomorphic. I wonder how to show this.

Thank you in advance for your help.

Answer 1

Here is one way to prove it. Let $(\varphi_t)$ be the one-parameter group generated by $X$. For fixed $t$, let $\psi_n$ be the approximation to $\varphi_t$ given by the $n$-step Euler method. Since $X$ is holomorphic, $\psi_n$ is holomorphic for each $n$. As $n\to\infty$, $\psi_n$ converges to $\varphi_t$ locally uniformly. Thus, $\varphi_t$ is also holomorphic.