# 微积分网课代修|函数代写Function theory代考|ISC5473 Conformality and Invariance

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|Conformality and Invariance

Conformal mappings are characterized by the fact that they infinitesimally (i) preserve angles, and (ii) preserve length (up to a scalar factor). It is worthwhile to picture the matter in the following manner: Let $f$ be holomorphic on the open set $U \subseteq \mathbb{C}$. Fix a point $P \in U$. Write $f=u+i v$ as usual. Thus we may write the mapping $f$ as $(x, y) \mapsto(u, v)$. Then the (real) Jacobian matrix of the mapping is

$$J(P)=\left(\begin{array}{ll} u_{x}(P) & u_{y}(P) \ v_{x}(P) & v_{y}(P) \end{array}\right)$$
where subscripts denote derivatives. We may use the Cauchy-Riemann equations to rewrite this matrix as
$$J(P)=\left(\begin{array}{cc} u_{x}(P) & u_{y}(P) \ -u_{y}(P) & u_{x}(P) \end{array}\right)$$
Factoring out a numerical coefficient, we finally write this two-dimensional derivative as
\begin{aligned} J(P) &=\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}} \cdot\left(\begin{array}{ll} \frac{u_{x}(P)}{\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}}} & \frac{u_{y}(P)}{\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}}} \ \frac{-u_{y}(P)}{\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}}} & \frac{u_{x}(P)}{\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}}} \end{array}\right) \ & \equiv h(P) \cdot \mathcal{J}(P) . \end{aligned}

(i) All rotations $\rho_{\lambda}: z \mapsto e^{i \lambda} \cdot z, 0 \leq \lambda<2 \pi$;
(ii) All Möbius transformations $\varphi_{a}: z \mapsto[z-a] /[1-\bar{a} z]$, $a \in \mathbb{C},|a|<1$
(iii) All compositions of mappings of type (i) and (ii).

## 微积分网课代修|偏微分方程代写Partial Differential Equation代考|Bergman’s Construction

Stefan Bergman created a device for equipping virtually any planar domain with an invariant metric that is analogous to the Poincaré metric on the disk. ${ }^{1}$ Some tracts call this new metric the Poincaré-Bergman metric, though it is more commonly called just the Bergman metric. In order to construct the Bergman metric we must first construct the Bergman kernel. For that we need just a little Hilbert space theory (see [RUD2], for example).

A domain in $\mathbb{C}$ is a connected open set. Fix a domain $\Omega \subseteq \mathbb{C}$, and define
$$A^{2}(\Omega)=\left{f \text { holomorphic on } \Omega: \int_{\Omega}|f(z)|^{2} d A(z)<\infty\right} \subseteq L^{2}(\Omega) .$$
Here $d A$ is an ordinary two-dimensional area measure. Of course $A^{2}(\Omega)$ is a complex linear space, called the Bergman space. The norm on $A^{2}(\Omega)$ is given by
$$|f|_{A^{2}(\Omega)}=\left[\int_{\Omega}|f(z)|^{2} d A(z)\right]^{1 / 2}$$
We define an inner product on $A^{2}(\Omega)$ by
$$\langle f, g\rangle=\int_{\Omega} f(z) \overline{g(z)} d A(z) .$$
The next technical lemma will be the key to our analysis of the space $A^{2}$.

## 微积分网课代修|偏微分方程代写Partial Differential Equation代 考 Conformality and Invariance

(实) 雅可比矩阵是
$$J(P)=\left(u_{x}(P) \quad u_{y}(P) v_{x}(P) \quad v_{y}(P)\right)$$

$$J(P)=\left(u_{x}(P) \quad u_{y}(P)-u_{y}(P) \quad u_{x}(P)\right)$$

$$J(P)=\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}} \cdot\left(\frac{u_{x}(P)}{\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}}} \quad \frac{u_{y}(P)}{\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}}} \frac{-u_{y}(P)}{\sqrt{u_{x}(P)^{2}+u_{y}(P)^{2}}} \frac{u_{x}(P)}{\sqrt{u_{x}(P)^{2}+u_{y}(F}}\right.$$
(i) 所有轮换 $\rho_{\lambda}: z \mapsto e^{i \lambda} \cdot z, 0 \leq \lambda<2 \pi$;
(ii) 所有莫比乌斯变换 $\varphi_{a}: z \mapsto[z-a] /[1-\bar{a} z], a \in \mathbb{C},|a|<1$
(iii) (i) 和 (ii) 类型映射的所有组合。

## 微积分网课代修偏微分方程代写Partial Differential Equation代考|Bergman’s Construction

Stefan Bergman 创建了一种设备，用于为几平任何平面域配备一个不变的度量，该度

$$|f|{A^{2}(\Omega)}=\left[\int{\Omega}|f(z)|^{2} d A(z)\right]^{1 / 2}$$

$$\langle f, g\rangle=\int_{\Omega} f(z) \overline{g(z)} d A(z)$$