Convergence and Holomorphicity of Series in Reinhardt Domains within Complex Analysis
Let $D \in \mathbb{C^n}$ be a Reinhardt domain and $f \in \mathcal{O}(D)$. For every $\alpha \in \mathbb{Z}^n$ define \[f_\alpha : D \to \mathbb{C}, z \mapsto \left(\frac{1}{2\pi i}\right)^{n}\int\limits_{\mathbb{T}^n}\frac{f(\zeta \cdot z)}{\zeta^{\alpha +1}}d\zeta (1.9) \] where $\zeta \cdot z = (\zeta_1z_1, ..., \zeta_nz_n).$ Then $f_\alpha \in \mathcal{O}(D)$ and the series $\sum\limits_{\alpha \in \mathbb{Z}^n}f_{\alpha}(z)$ converges compactly on $D$ towards $f$.
Prove the following.
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to 50% commission on every question your affiliated users ask or answer.
- accepted
- 177 views
- $40.00
Related Questions
- Fixed points of analytic complex functions on unit disk $\mathbb{D}$
- Calculating the residues at the poles of $f(z) = \frac{\tan(z) }{z^2 + z +1} $
- Given the complex polynomial equation $z^{n} -1=0$ for n=2, 3, 4, what are the corresponding roots?
- Two exercises in complex analysis
- Complex Numbers Assignment 2
- Complex Numbers
- Advanced Modeling Scenario
- Prove that $\int _0^{\infty} \frac{1}{1+x^{2n}}dx=\frac{\pi}{2n}\csc (\frac{\pi}{2n})$