数学代写|黎曼曲面代写Riemann surface代考|The Weierstraß ℘-function

Doug I. Jones

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数学代写|黎曼曲面代写Riemann surface代考|The Weierstraß ℘-function

We assume a lattice $L=\mathbb{Z} \omega_1+\mathbb{Z} \omega_2 \subset \mathbb{C}$ has been chosen, with $\omega_1, \omega_2 \neq 0, \omega_1 / \omega_2 \notin \mathbb{R}$.
We know that an elliptic function cannot have a single, simple pole in the period paralellogram. So the simplest assignment of principal parts is a double pole at $u=0$. This leads to the $\wp$-function of Weierstraß.
9.1 Theorem/Definition: The Weierstraß $\wp$-function is the sum of the series
$$\wp(u)=\frac{1}{u^2}+\sum_{\omega \in L^*}\left[\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right]$$
which converges to an elliptic function; the convergence is uniform on any compact subset $K \subset \mathbb{C}$, once the terms with poles in $K$ are set aside.

Proof of convergence: For a compact $K \subset \mathbb{C}$, only finitely many terms will have poles in $K$. If the others converge uniformly, as we claim, holomorphy of their sum, and hence meromorphy of the entire series is a consequence of Morera’s Theorem. Now, the individual terms are estimated by
$$\left|\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right|<\frac{|u|^2+2|u||\omega|}{|\omega|^2|u-\omega|^2}=\frac{|u|^2}{|\omega|^2|u-\omega|^2}+2 \frac{|u|}{|\omega||u-\omega|^2}$$ and we have estimates $|u-\omega|>a^{-1}|\omega|,|u|<b$ for $u \in K$ and $\omega \in L \backslash K$; so
$$\left|\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right|<\frac{a^2 b^2}{|\omega|^4}+\frac{2 a^2 b}{|\omega|^3}$$ and the series on the right converges. (Proof: estimate by comparing with $\iint\left(x^2+y^2\right)^{-k}: d x d y$, with $k=\frac{3}{2}$ and $k=2$. Alternatively, sum the series on parallelograms centered at the origin, and show that the sum over the boundary of each parallelogram decays at least quadratically with respect to the size of the parallelogram.)

数学代写|黎曼曲面代写Riemann surface代考|Corollary

9.2 Corollary: The series for $\wp(u)$ can be differentiated term by term and gives
$$\wp^{\prime}(u)=-2 \sum_{\omega \in L} \frac{1}{(u-\omega)^3},$$
an elliptic function with a triple pole at 0 .
9.3 Proposition: $\wp$ is even, that is, $\wp(u)=\wp(-u) ; \wp^{\prime}$ is odd, that is, $\wp^{\prime}(u)=-\wp^{\prime}(-u)$.
Proof: Clear from the series expansion.
First proof of periodicity of $\wp$ : The derivative $\wp^{\prime}(u)$ is evidently periodic, as a period shift in $u$ can be reabsorbed by the lattice. But then,
$$\frac{d}{d u}\left(\wp(u)-\wp\left(u+\omega_i\right)\right)=\wp^{\prime}(u)-\wp^{\prime}\left(u+\omega_i\right)=0,$$
so $\wp(u)-\wp\left(u+\omega_i\right)$ is constant. Setting $u=-\omega_i / 2$ and using the parity of $\wp$, we see that the constant is zero.
Second proof of periodicity: As a consequence of convergence,
\begin{aligned} \wp\left(u+\omega_1\right) & =\frac{1}{\left(u+\omega_1\right)^2}+\sum_{\omega \in L^}\left[\frac{1}{\left(u+\omega_1-\omega\right)^2}-\frac{1}{\omega^2}\right] \ & =\frac{1}{\left(u+\omega_1\right)^2}+\left[\frac{1}{u^2}-\frac{1}{\omega_1^2}\right]+\sum_{\substack{\omega \in L^ \ \omega \neq \omega_1}}\left[\frac{1}{\left(u+\omega_1-\omega\right)^2}-\frac{1}{\omega^2}\right] \ & =\frac{1}{u^2}+\frac{1}{\left(u+\omega_1\right)^2}-\frac{1}{\omega_1^2}+\sum_{\substack{\omega \in L^* \ \omega \neq-\omega_1}}\left[\frac{1}{(u-\omega)^2}-\frac{1}{\left(\omega+\omega_1\right)^2}\right] \ & =\frac{1}{u^2}+\left[\frac{1}{\left(u+\omega_1\right)^2}-\frac{1}{\omega_1^2}\right]+\sum_{\substack{\omega \in L^* \ \omega \neq-\omega_1}}\left[\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right]+\sum_{\substack{\omega \in L^* \ \omega \neq-\omega_1}}\left[\frac{1}{\omega^2}-\frac{1}{\left(\omega+\omega_1\right)^2}\right] \ & =\wp(u)+\sum_{\substack{\omega \in L^* \ \omega \neq-\omega_1}}\left[\frac{1}{\omega^2}-\frac{1}{\left(\omega+\omega_1\right)^2}\right] . \end{aligned}

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数学代写|黎曼曲面代写Riemann surface代考|The Weierstraß ℘-function

9.1定理/定义:Weierstraß $\wp$ -函数是级数的和
$$\wp(u)=\frac{1}{u^2}+\sum_{\omega \in L^*}\left[\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right]$$

$$\left|\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right|<\frac{|u|^2+2|u||\omega|}{|\omega|^2|u-\omega|^2}=\frac{|u|^2}{|\omega|^2|u-\omega|^2}+2 \frac{|u|}{|\omega||u-\omega|^2}$$，我们对$u \in K$和$\omega \in L \backslash K$有估算$|u-\omega|>a^{-1}|\omega|,|u|<b$;所以
$$\left|\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right|<\frac{a^2 b^2}{|\omega|^4}+\frac{2 a^2 b}{|\omega|^3}$$右边的级数收敛。(证明:通过与$\iint\left(x^2+y^2\right)^{-k}: d x d y$、$k=\frac{3}{2}$和$k=2$的比较估算。或者，将以原点为中心的平行四边形的级数相加，并表明每个平行四边形边界上的总和至少与平行四边形的大小成二次衰减。)

数学代写|黎曼曲面代写Riemann surface代考|Corollary

9.2推论:$\wp(u)$的级数可逐项求导，并给出
$$\wp^{\prime}(u)=-2 \sum_{\omega \in L} \frac{1}{(u-\omega)^3},$$

9.3命题:$\wp$是偶数，即$\wp(u)=\wp(-u) ; \wp^{\prime}$是奇数，即$\wp^{\prime}(u)=-\wp^{\prime}(-u)$。

$$\frac{d}{d u}\left(\wp(u)-\wp\left(u+\omega_i\right)\right)=\wp^{\prime}(u)-\wp^{\prime}\left(u+\omega_i\right)=0,$$

\begin{aligned} \wp\left(u+\omega_1\right) & =\frac{1}{\left(u+\omega_1\right)^2}+\sum_{\omega \in L^}\left[\frac{1}{\left(u+\omega_1-\omega\right)^2}-\frac{1}{\omega^2}\right] \ & =\frac{1}{\left(u+\omega_1\right)^2}+\left[\frac{1}{u^2}-\frac{1}{\omega_1^2}\right]+\sum_{\substack{\omega \in L^ \ \omega \neq \omega_1}}\left[\frac{1}{\left(u+\omega_1-\omega\right)^2}-\frac{1}{\omega^2}\right] \ & =\frac{1}{u^2}+\frac{1}{\left(u+\omega_1\right)^2}-\frac{1}{\omega_1^2}+\sum_{\substack{\omega \in L^* \ \omega \neq-\omega_1}}\left[\frac{1}{(u-\omega)^2}-\frac{1}{\left(\omega+\omega_1\right)^2}\right] \ & =\frac{1}{u^2}+\left[\frac{1}{\left(u+\omega_1\right)^2}-\frac{1}{\omega_1^2}\right]+\sum_{\substack{\omega \in L^* \ \omega \neq-\omega_1}}\left[\frac{1}{(u-\omega)^2}-\frac{1}{\omega^2}\right]+\sum_{\substack{\omega \in L^* \ \omega \neq-\omega_1}}\left[\frac{1}{\omega^2}-\frac{1}{\left(\omega+\omega_1\right)^2}\right] \ & =\wp(u)+\sum_{\substack{\omega \in L^* \ \omega \neq-\omega_1}}\left[\frac{1}{\omega^2}-\frac{1}{\left(\omega+\omega_1\right)^2}\right] . \end{aligned}

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