I am stuck at the following question : Let $f$ be a holomorphic function on $\mathbb{C}^n \setminus \{(z_1, z_2, \ldots, z_n) | z_1=z_2=0\}$. Show that $f$ can be extended to a holomorphic function on $\mathbb{C}^n$. I think that I have to somehow use the Riemann extension theorem here. But I am not being able to […]

In one variable complex theory, we have the result that zeroes of a non-zero analytic function are isolated. In several variable theory, this result does not hold. I read it somewhere that this fact can be proved using Hurwitz theorem. If anyone can help me with this.

Let $\Omega$ be a domain in $\mathbb{C}^n.$ Consider the Banach algebra $A(\Omega):=\mathcal{C}({\overline{\Omega}})\cap\mathcal{O}(\Omega).$ Denote the Bergman-Shilov boundary of $A(\Omega)$ by $\partial_S(\Omega)$ . From the very definiton of peak points we know that every peak point belongs to $\partial_S(\Omega).$ Now the examples that I know the set of peak points coincides with the $\partial_S(\Omega).$ My question here […]

Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that $$K_\Omega(z,\zeta) = \sum_{n=1}^\infty \varphi_k(z) \overline{\varphi_k(\zeta)},\qquad(z,\zeta\in\Omega)$$ where $\{\varphi_k\}_{k=1}^\infty$ is any orthonormal basis of the Bergman space $A^2(\Omega)$ of Lebesgue square-integrable holomorphic functions on $\Omega$. This series representation converges at least pointwise, since the […]

I would like a HINT for this: Exhibit a two variable power series whose convergence domain is the unit ball $\{(z,w):|z|^2+|w|^2 < 1\}$. ($z$ and $w$ are complex numbers.) I think that it cannot be of the form $\sum P(z,w)^n$ where $P(z,w)$ is a polynomial. But I’m out of ideas. Thank you.

Is there an extension of the residue theorem to multivariate complex functions? Say you have a function of $n$ complex variables $s_{n}$ and you wish to integrate it over some region in $\mathbb{C}^{n}$. Can you exploit the singularities of the function as you would in the single variable case to evaluate the integral? For example […]

Let $\Omega\subset \mathbb{C}^n$ be an open unit ball, $f:\Omega \to\mathbb{C}$ is a bounded function. For $a \in \mathbb{C}^n$, define $$ \Omega_{j,a}=\{z\in\mathbb{C}:(a_1,…a_{j-1},z,a_{j+1},…,a_n)\in \Omega\}$$ If $f(a_1,…a_{j-1},z,a_{j+1},…,a_n)$is holomorphic on $\Omega_{j,a}$ for each$a\in\mathbb{C}^n$. Show that $f:\Omega \to\mathbb{C}$ is continuous. Moreover, $f$ is also holomorphic. (Definition:$f:\Omega \to\mathbb{C}$ is said to be holomorphic if for any $a\in \Omega$ there exists a […]

Suppose $D=\Delta^n(a,r)=\Delta(a_1,r_1)\times \ldots \times \Delta(a_n,r_n) \subset \mathbb{C}^n$ and $\Gamma =\partial \circ \Delta^n(a,r)=\left \{ z=(z_1, \ldots , z_n)\in \mathbb{C}^n:|z_j-a_j|=r_j,~ j=\overline{1,n} \right \}$. Let $f \in \mathcal{H}(D) \cap \mathcal{C}(\overline{D})$. Prove that: $\sup_{z \in \overline{D}} |f(z)|=\sup_{z \in \Gamma} |f(z)|$ I need your help. Thanks.

Why is the zero set in $\mathbb{C}$ of a polynomial $f(x,y)$ in two complex variables always non-discrete (no zero of $f$ is isolated)?

I am looking for a generalization of the Cauchy integral theorem. I know that there are generalizations of the Cauchy integral formula (eg the Bochner-Martinelli formula), but I do not know if this simplifies as I would hope. In single-variable complex analysis, we have the Cauchy integral theorem: $$ \oint_{\gamma}f(z) = 0 $$ if $f(z)$ […]

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