## 数学代写|黎曼曲面代写Riemann surface代考|MATH501

2023年2月1日

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## 数学代写|黎曼曲面代写Riemann surface代考|COMPLEMENTS AND EXAMPLES

In this section we will review what has been proved, make some further comments, and give some examples.
Review
The transport matrix $m(Q, t)$ is given by the infinite sum of integrals
$$m(Q, t)=\sum_n m_n(Q, t), \quad m_n(Q, t)=\sum_{|I|=n} \int_{\beta_I} b e^{t g} .$$
The Laplace transform is again an infinite sum
$$f(\zeta)=\sum_n f_n(\zeta), \quad f_n(\zeta)=\sum_{|I|=n} \int_{\beta_I} \frac{b}{g-\zeta} .$$
We have shown in $\S \S 4-10$ that this infinite sum satisfies the conditions (2.5.0)(2.5.4), and furthermore that if $B$ was multiplied by a generic number $\chi$ then it satisfies (2.5.5). Proposition $2.5$ shows that $f(\zeta)$ has an extension with locally finite branching and quasi-regular singularities. As noted in the remark following $2.5$, the sum of the expansions $\hat{f}{n, \text {,ing }}$ converges formally to the expansion $\hat{f}{\text {aing }}$ at any singularity. By $2.3$, the transport matrix $m(Q, t)$ has an asymptotic expansion, which is the formal sum of the expansions for $m_n(Q, t)$. If $B$ is multiplied by a generic number, then $2.5$ shows that $f(\zeta)$ has faithful expansions, so the asymptotic expansion for $m(Q, t)$ is nonzero. This is the proof of Theorem 1 (and Variant 1.1).

## 数学代写|黎曼曲面代写Riemann surface代考|Here X is a finite generic parameter

Proposition 11.3 Suppose $\Phi(t)$ is a polynomial in derivatives of the matrix coefficients $m_{i j}(Q, t)$, with coefficients which are polynomials in $t$. Let $f(\zeta)$ be the Laplace transform of $\Phi(t)$. Then $f(\zeta)$ has an analytic continuation with locally finite branching and quasi-regular singularities. If $\chi$ is chosen generically, then $f(\zeta)$ has faithful expansions. Consequently $\Phi(t)$ has an asymptotic expansion as $t \rightarrow \infty$ in any given direction. The asymptotic expansion is nonzero if $\chi$ is generic and $\Phi(t)$ is not identically zero.

Proof: The Laplace transform of a polynomial in $t$ is a meromorphic function with some poles at the origin only. Similarly, taking the derivative of a function corresponds to multiplying its Laplace transform by $\zeta$ (and subtracting off the appropriate constant to maintain vanishing at infinity). This preserves the conditions 2.5. By Lemma 11.1, the Laplace transform $f(\zeta)$ of the polynomial $\Phi(t)$ will be a sum of convolutions of Laplace transforms of polynomials in $t$, and Laplace transforms of matrix coefficients $m_{i j}(t)$ (or their derivatives). Let $\mathbf{k}$ denote the subfield generated over $\mathbf{Q}(\Gamma)$ by all coefficients in the case $\chi=1$. Then assume that $\chi$ is transcendentally independent of $\mathbf{k}$, and let $H_n=\chi^n \mathbf{k} \subset$ C. The vector spaces $H_n$ are independent over $\mathbf{Q}(\Gamma), H_m H_n \subset H_{n+m}$, and the coefficients of the expansions satisfy condition (2.5.5) with respect to these $H_n$ (because if $f(\zeta)$ is a Laplace transform of a matrix coefficient, then the integrands of the terms $f_n(\zeta)$ are homogeneous of degree $n$ in the matrix $B$ ). Apply Propositions $2.5$ and $11.2$ to obtain the conclusions.

Remark: The same statement and proof hold if the coefficients of $\Phi$ are functions whose Laplace transforms have locally finite regular singularities.
Remark: One would like to show a strong transcendence statement, namely that the matrix coefficients $m_{i j}(Q, t)$ do not satisfy any differential equation, even with exponential functions (or functions whose Laplace transforms have finitely many regular singularities) as coefficients. The analytic continuations of the Laplace transforms that we have obtained should be helpful here. The basic problem remains to figure out a good method of calculating the locations of the singularities and the coefficients of the quasi-regular expansions.

# 黎曼曲面代写

## 数学代写|黎曼曲面代写Riemann surface代考|COMPLEMENTS AND EXAMPLES

$$m(Q, t)=\sum_n m_n(Q, t), \quad m_n(Q, t)=\sum_{|I|=n} \int_{\beta_I} b e^{t g}$$

$$f(\zeta)=\sum_n f_n(\zeta), \quad f_n(\zeta)=\sum_{|I|=n} \int_{\beta_I} \frac{b}{g-\zeta}$$

(或拉普拉斯变换具有有限多个正则奇点的函数) 作为 系数。我们获得的拉普拉斯变换的解析延拓在这里应该 会有帮助。基本问题仍然是找出计算奇点位置和准正则 展开系数的好方法。

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