Let $\Omega$ be a domain in $\mathbb{C}$. A function $f:\Omega\to\mathbb{C}$ is said to be real analytic on $\Omega$ if for each point $z_{0}\in\Omega$, there exists a neighbourhood $N_{z_{0}}$ where $f(z)=\sum_{i,j=0}^{\infty} a_{ij}z^{i}\bar{z}^j$ for all $z\in N_{z_{0}}$. My question is, can there exists a sesqui analytic function $F:\Omega\times\Omega\to\mathbb{C}$ such that $f=F|_{\Delta}$, where $\Delta=\{(z,z)|z\in\Omega\}$?
Can a real analytic function be realized as a restriction of a sesquianalytic function to the digonal
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complex-analysis
several-complex-variables