### 数学代写|黎曼曲面代写Riemann surface代考|KMA152

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## 数学代写|黎曼曲面代写Riemann surface代考|ORDINARY DIFFERENTIAL EQUATIONS ON A PdEMANN SURFACE

Let $S$ be a compact Riemann surface. We will consider systems of first order ordinary differential equations on $S$
$$(d-t A-B) m=0$$
where $A$ and $B$ are $k \times k$ matrices of holomorphic one-forms on $S, t$ is a complex parameter, and $m$ is a column vector or $k \times k$ matrix of functions. We make the following assumption:
$A$ is a diagonal matrix with one-forms $a_1, \ldots, a_k$ along the diagonal. The diagonal entries of $B$ are equal to zero.

The solutions of the system of differential equations are multivalued holomorphic functions on $S$, so it is more convenient to introduce the universal cover $Z=\tilde{S}$. This is complex analytically equivalent to a domain in the complex plane, and it is sometimes useful to keep such an embedding in mind.

Fix a base point $P$ in $Z$ (lying above a base point which we also denote by $P$ in $S$ ). There is a unique matrix valued solution $m(z)$ defined for $z \in Z$, specified by initial conditions $m(P)=I$. For any point $Q$ on $Z$, the value $m(Q)$ is well defined. It depends on the parameter $t$, so we obtain an entire matrix valued function $m(t)=m(Q, t)$ of the complex variable $t$.

Our aim is to investigate the behavior of $m(t)$ as $t \rightarrow \infty$. We can state a theorem which is essentially the main result. Restrict to positive real values of t. Recall that an asymptotic expansion for $m(t)$ is an expression
$$m(t) \sim \sum_{i=1}^{\Gamma} \sum_{j=J}^{\infty} \sum_{k=0}^{K(j)} c_{i j k} e^{\lambda_i t} t^{-\frac{1}{N}}(\log t)^k$$
where the real parts of the exponents are equal-say $\Re \lambda_i=\xi$ for all $i$, such that for each $M$ there is a $y(M)$ and a constant $C(M)$ such that for $t \geq y(M)$,
$$\left|m(t)-\sum_{i=1}^r \sum_{j=J}^{N M} \sum_{k=0}^{K(j)} c_{i j k} e^{\lambda, t} t^{-\frac{1}{N}}(\log t)^k\right| \leq C(M) e^{\xi t} t^{-M} .$$
Call the numbers $\lambda_i$ the complex exponents of the expansion, and the number $\xi$ the real exponent or just the exponent.

## 数学代写|黎曼曲面代写Riemann surface代考|LAPLACE TRANSFORM, ASYMPTOTIC EXPANSIONS

Classically, the method of the stationary phase (or steepest descent) provided asymptotic expansions for integrals such as
$$\int f(z) e^{-t x^2} d z .$$
In this paper we are interested in obtaining asymptotic expansions for more general integrals such as
$$m(t)=\int_\eta b e^{t g},$$
where $g$ is a holomorphic function on a complex manifold, $b$ is a holomorphic differential form of top degree, and $\eta$ is a cycle in homology or relative homology (of real dimension equal to the complex dimension of the manifold). Instead of applying the method of stationary phase directly to such an integral, it will be more useful to take the Laplace transform first. The Laplace transform keeps lower order information which is lost upon going to the asymptotic expansion. If several such integrals are added together and their asymptotic expansions cancel, then an asymptotic expansion at lower exponent can be recovered from the sum of the Laplace transforms.

Suppose that $m(t)$ is an entire holomorphic function of order $\leq 1$. This means that there is a bound
$$|m(t)| \leq C e^{a|t|} .$$
The Laplace transform of $m$ is defined to be the integral
$$f(\zeta)=\int_0^{\infty} m(t) e^{-\zeta t} d t .$$
The integration is taken along a direction in which the integrand is rapidly decreasing. $f(\zeta)$ is defined and holomorphic for $|\zeta|>a$, and it vanishes at $\infty$. Conversely the function $m(t)$ can be recovered as the inverse Laplace transform
$$m(t)=\frac{1}{2 \pi i} \oint f(\zeta) e^{\zeta t} d \zeta .$$
Here the path of integration is a large circle running once counterclockwise around the annulus $|\zeta|>a$.

# 黎曼曲面代考

## 数学代写|黎曼曲面代写Riemann surface代考|ORDINARY DIFFERENTIAL EQUATIONS ON A PdEMANN SURFACE

$$(d-t A-B) m=0$$

$A$ 是具有一种形式的对角矩阵 $a_1, \ldots, a_k$ 沿着对角线。的对角线条目 $B$ 等于零。

$$m(t) \sim \sum_{i=1}^{\Gamma} \sum_{j=J}^{\infty} \sum_{k=0}^{K(j)} c_{i j k} e^{\lambda_i t} t^{-\frac{1}{N}}(\log t)^k$$

$$\left|m(t)-\sum_{i=1}^r \sum_{j=J}^{N M} \sum_{k=0}^{K(j)} c_{i j k} e^{\lambda, t} t^{-\frac{1}{N}}(\log t)^k\right| \leq C(M) e^{\xi t} t^{-M}$$

## 数学代写|黎曼曲面代写Riemann surface代考|LAPLACE TRANSFORM, ASYMPTOTIC EXPANSIONS

$$\int f(z) e^{-t x^2} d z$$

$$m(t)=\int_\eta b e^{t g},$$

$$|m(t)| \leq C e^{a|t|} .$$

$$f(\zeta)=\int_0^{\infty} m(t) e^{-\zeta t} d t$$

$$m(t)=\frac{1}{2 \pi i} \oint f(\zeta) e^{\zeta t} d \zeta .$$

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