Posted on Categories:Multivariable Complex Analysis, 多复变函数论, 数学代写

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## 数学代写|多复变函数论代考Multivariable Complex Analysis代写|Cauchy integral formula for a polydisc

For functions $f$ that are holomorphic on a closed polydisc $\bar{\Delta}(a, r)$, there is an integral representation of Cauchy which extends the well-known one-variable formula. We will actually assume a little less than holomorphy:

Theorem 1.3.1. Let $f(z)=f\left(z_1, \ldots, z_n\right)$ be continuous on $\Omega \subset \mathbb{C}^n$ and differentiable in the complex sense with respect to each of the variables $z_j$ separately. Then for every closed polydisc $\bar{\Delta}(a, r) \subset \Omega$
$$f(z)=\frac{1}{(2 \pi i)^n} \int_{T(a, r)} \frac{f(\zeta)}{\left(\zeta_1-z_1\right) \ldots\left(\zeta_n-z_n\right)} d \zeta_1 \ldots d \zeta_n, \quad \forall z \in \Delta(a, r)$$
where $T(a, r)$ is the torus $C\left(a_1, r_1\right) \times \ldots \times C\left(a_n, r_n\right)$, with positive orientation of the circles $C\left(a_j, r_j\right)$.
Proof. We write out a proof for $n=2$. In the first part we only use the complex differentiability of $f$ with respect to each variable $z_j$, not the continuity of $f$.

Fix $z$ in $\Delta(a, r)=\Delta_1\left(a_1, r_1\right) \times \Delta_1\left(a_2, r_2\right)$ where $\bar{\Delta}(a, r) \subset \Omega$. Then $g(w)=f\left(w, z_2\right)$ has a complex derivative with respect to $w$ throughout a neighbourhood of the closed disc $\bar{\Delta}1\left(a_1, r_1\right)$ in $\mathbb{C}$. The one-variable Cauchy integral formula thus gives $$f\left(z_1, z_2\right)=g\left(z_1\right)=\frac{1}{2 \pi i} \int{C\left(a_1, r_1\right)} \frac{g(w)}{w-z_1} d w=\frac{1}{2 \pi i} \int_{C\left(a_1, r_1\right)} \frac{f\left(\zeta_1, z_2\right)}{\zeta_1-z_1} d \zeta_1 .$$

## 数学代写|多复变函数论代考Multivariable Complex Analysis代写|Multiple power series

The general power series in $\mathbb{C}^n$ with center $a$ has the form
$$\sum_{\alpha_1 \geq 0, \ldots, \alpha_n \geq 0} c_{\alpha_1 \ldots \alpha_n}\left(z_1-a_1\right)^{\alpha_1} \ldots\left(z_n-a_n\right)^{\alpha_n} .$$
Here the $\alpha_j$ ‘s are nonnegative integers and the $c$ ‘s are complex constants. We will see that multiple power series have properties similar to those of power series in one complex variable.

Before we start it is convenient to introduce abbreviated notation. We write $\alpha$ for the multi-index or ordered $n$-tuple $\left(\alpha_1, \ldots, \alpha_n\right)$ of integers. Such $n$-tuples are added in the usual way; the inequality $\alpha \geq \beta$ will mean $\alpha_j \geq \beta_j, \forall j$. In the case $\alpha \geq 0$ [that is, $\alpha_j \geq 0, \forall j$ ], we also write
$$\alpha \in \mathbb{C}0^n, \quad \alpha !=\alpha{1} ! \ldots \alpha_{n} !, \quad|\alpha|=\alpha_1+\ldots+\alpha_n \quad \text { (height of } \alpha \text { ) } .$$
One sets
$$z_1^{\alpha_1} \ldots z_n^{\alpha_n}=z^\alpha, \quad\left(z_1-a_1\right)^{\alpha_1} \ldots\left(z_n-a_n\right)^{\alpha_n}=(z-a)^\alpha,$$
so that the multiple sum (1.4.1) becomes simply
$$\sum_{\alpha \geq 0} c_\alpha(z-a)^\alpha .$$

# 多复变函数论代考

## 数学代写|多复变函数论代考Multivariable Complex Analysis代写|Cauchy integral formula for a polydisc

$$f(z)=\frac{1}{(2 \pi i)^n} \int_{T(a, r)} \frac{f(\zeta)}{\left(\zeta_1-z_1\right) \ldots\left(\zeta_n-z_n\right)} d \zeta_1 \ldots d \zeta_n, \quad \forall z \in \Delta(a, r)$$

$$f\left(z_1, z_2\right)=g\left(z_1\right)=\frac{1}{2 \pi i} \int C\left(a_1, r_1\right) \frac{g(w)}{w-z_1} d w=\frac{1}{2 \pi i} \int_{C\left(a_1, r_1\right)} \frac{f\left(\zeta_1, z_2\right)}{\zeta_1-z_1} d \zeta_1 .$$

## 数学代写|多复变函数论代考Multivariable Complex Analysis代写|Multiple power series

$$\sum_{\alpha_1 \geq 0, \ldots, \alpha_n \geq 0} c_{\alpha_1 \ldots \alpha_n}\left(z_1-a_1\right)^{\alpha_1} \ldots\left(z_n-a_n\right)^{\alpha_n} .$$

$$\alpha \in \mathbb{C} 0^n, \quad \alpha !=\alpha 1 ! \ldots \alpha_{n} !, \quad|\alpha|=\alpha_1+\ldots+\alpha_n \quad \text { (height of } \alpha \text { ) . }$$

$$\sum_{\alpha \geq 0} c_\alpha(z-a)^\alpha .$$

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## MATLAB代写

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