Hallo,


Let $f: U \rightarrow \mathbb{R}$ be a analytic function, where $U \subset \mathbb{C}^{n}$ is a open set (paracompact, starshaped or convex i.e. sufficiently nice). Does there exist a function $\varphi : U \rightarrow \mathbb{R}$ such that $f + i \varphi$ is holomorphic? Is this possible?


hapchiu 



Constructing the imaginary part of a holomorphic function
Suppose $n=1$. Then, if $f=u+iv$ is holomorphic, $u,v$ are harmonic, i.e. $u\_{xx}+u\_{yy}=0$. Hence a necessary condition for $u$ to be the real part of a holomorphic function is that $u$ is harmonic. The converse is true if $U$ is simply connected: See Theorem 6.3 in <http://www.math.binghamton.edu/sabalka/teaching/09Spring375/Chapter6.pdf>. 


In the general case ($n \ge 1$) one has to consider *pluriharmonic* functions: A function $u(x\_1,y\_1,...,x\_n,y\_n): U \subseteq \mathbb{R}^{2n} \to \mathbb{R}$ is called pluriharmonic, if it is $C^2$ and satisfies $u\_{x\_jx\_k}+u\_{y\_jy\_k}=0$ for all $1 \le j,k\le n$. 


The real part of a holomorphic function $f: U \to \mathbb{C}$ is pluriharmonic and conversely, each pluriharmonic function is locally (in a disc) the real part of a holomorphic function. See Theorem 26 of <http://www.dms.umontreal.ca/~gauthier/6140.pdf>. 


