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Suppose I have a function $f: \mathbb C^d \rightarrow \mathbb C$, that i know in close form, and from which i want to bound the taylor coefficients around 0. For that, I am using the cauchy integral formula, and i want to be sure that i have it right.

I start by defining the polydisc $D_{\mathbf \rho} = \left\{\mathbf z \in \mathbb C^d: \; \lvert z_i \rvert < \rho_i\right\}$, such that no singularity of $f$ are in this disc.

Then i can say that, if i expand $f$ as: $$f(\mathbf z) = \sum\limits_{\mathbf i \in \mathbb N^d} f_{\mathbf i} \mathbf z^{\mathbf i},$$

then i have by the cauchy integral formula that :

$$\lvert a_{\mathbf i}\rvert \le (2\pi)^d \mathbf \rho^{-\mathbf k} \sup\limits_{\mathbf z \in \mathbf D} \lvert f(\mathbf z) \rvert$$

Q1: Is that statement correct ?

Q2: To define the corect polydisc, i'm trying to understand the notion of singularity of the function. If for a bivariate function i find a singularity at $\mathbb{C} \times \{z_2\}$ and another one at $\{z_1\} \times \mathbb{C}$, for both $\lvert z_1\rvert $ and $\lvert z_2\rvert$ greater than one, how should I define the greatest polydisc that works ?

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  • $\begingroup$ see here $\endgroup$
    – user111
    Dec 19, 2020 at 13:51

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