Problem 1.

Using the eigenfunctions $y_i(x)$ of the equation
$$y(x)=\lambda \int_0^1 e^{x s} y(s) d s,$$
approximate the following functions $f(x)$ for $x \in[0,1]$ in ten-term expansions of the form
$$f(x)=\sum_{i=1}^{10} \alpha_i y_i(x)$$
(a) $f(x)=x$
(b) $f(x)=\sin (\pi x)$
The eigenfunctions will need to be estimated by numerical approximation.

Problem 2.

Find the Fourier transformation of the Gaussian function $f(x)=e^{-x^2 / 2}$.

We get for our Gaussian function
\begin{aligned} F(\alpha) & =\int_{-\infty}^{\infty} e^{-x^2 / 2} e^{-i \alpha x} d x, \ & =\sqrt{2 \pi} e^{-\alpha^2 / 2} . \end{aligned}
The Gaussian has identical spatial and spectral localization.

## MATH0010 (Mathematical Methods 1)

$\begin{array}{ll}\text { Year: } & 2019-2020 \ \text { Code: } & \text { MATH0010 } \ \text { Old code: } & \text { MATH1401 } \ \text { Level: } & 4 \text { (UG) } \ \text { Normal student group (s): } & \text { UG: Year 1 Mathematics degrees } \ \text { Value: } & 15 \text { credits (= } 7.5 \text { ECTS credits) } \ \text { Term: } & 1 \ \text { Structure: } & 3 \text { hours lectures, 1 hour problem class per week. Small group tutori- } \ & \text { als. Weekly assessed coursework. } \ \text { Assessment: } & \text { The final weighted mark for the module is given by: } 85 \% \text { examination, } \ & 10 \% \text { coursework, } 5 \% \text { calculus test. The coursework mark is obtained } \ & \text { from exercise sheet marks, a test on vectors and the mid-sessional } \ & \text { examination result. In order to pass the module you must have at } \ & \text { least 40\% for both the examination and the final weighted mark and } \ & \text { must also pass the calculus test. } \ \text { A* in A-level Mathematics and Further Mathematics } \ \text { Normal Pre-requisites: } & \text { Prof R Halburd } \ \text { Lecturer: } & \text { Dr D Hewett }\end{array}$
Course Description and Objectives
The aim of the course is to bring students from a background of diverse A-level syllabuses to a uniform level of confidence and competence in vectors, complex numbers, calculus and differential equations. The course covers vectors, complex numbers, standard functions of a real variable, methods of integration, ordinary differential equations and probability. Each topic is given a formal treatment and illustrated by examples of varying degrees of difficulty.

There are two tests which are part of this course: a vectors test and a calculus test. The vectors test takes place around weeks 5 or 6 of term 1 and counts as part of the course work. This test is offered once. On the other hand, the calculus test is offered several times during the term, consisting of 10 basic calculational questions; you must get at least 9 correct answers to pass. It is necessary to pass this test, which you may attempt several times, in order to pass the module.

Fourier analysis代写

## 离散数学代写

Partial Differential Equations代写可以参考一份偏微分方程midterm答案解析