## MAS342 Applicable Analysis

Semester 2, 2017/18 | 10 Credits | ||||

Lecturer: | Dr Mary Hart | Reading List | |||

Aims | Outcomes | Teaching Methods | Assessment | Full Syllabus |

Over the years mathematical tools have been developed to solve practical problems which have arisen naturally in the course of research. Many of these problems involve the evaluation of integrals or the solution of differential equations and so are essentially concerned with calculus. This is a course made up of topics which have numerous applications and is ideal for those who can cope with calculus and enjoy it.

The aim of this module is to develop the theory of a number of analytical tools in such a way as to acquaint the students with the underlying theory and to teach them the capabilities and limitations of the methods. The course will include plenty of examples so that students learn to use the tools correctly. Topics covered are improper integrals, Gamma and Beta functions and the theory of Laplace transforms. They are used to evaluate integrals and to solve ordinary and partial differential equations. As some students will reach the third year without meeting differential equations, an introductory section on differential equations is included in the course.
**Prerequisites:** MAS211 (Advanced Calculus and Linear Algebra) ; MAS332 (Complex Analysis) desirable but not essential.

No other modules have this module as a prerequisite.

## Outline syllabus

- Improper integrals of the first and second kind
- Change of order in repeated integrals of the form ∫
_{c}^{d}∫_{a}^{∞}f (x, y) dx dy; differentiation under the integral sign for ∫_{a}^{∞}f (x, t) dt - Gamma and Beta functions and the relationship between them
- Applications of Gamma and Beta functions
- Laplace transforms
- The convolution of two functions and its Laplace transform
- Applications of Laplace transforms to the evaluation of integrals and the solution of ordinary and partial differential equations

## Aims

- To introduce students to some topics which are analytical in nature and are widely applicable
- To train the students to be able to use the Gamma and Beta functions and Laplace transforms correctly to solve a variety of problems

## Learning outcomes

- test Improper integrals for convergence;
- use Gamma and Beta functions to evaluate integrals;
- use Laplace transforms to evaluate integrals;
- use Laplace transforms to solve ordinary differential equations;
- use Laplace transforms to solve partial differential equations;
- quote the statement of theorems (including all the conditions which guarantee their validity) and demonstrate that they can use these results correctly to solve problems (including checking conditions before applying the conclusion of the theorem).

## Teaching methods

There are two lectures per week. All the material for the course is on "MOLE2" so students can spend their time in lectures thinking about the work rather than just copying down notes.

At times extra problems are incorporated into the lecture. These extra problems and their solutions are then added to the relevant folder in "MOLE2". There are Exercise Sheets on all parts of the course and students are asked to submit solutions to a number of these at about fortnightly intervals. Detailed solutions for any exercises sheet are made available to view after the problems from the exercise sheet have been handed in. Students are also advised that they would be wise to attempt the remaining problems also. As there are no formal tutorials, several office hours a week are allocated so that students can consult the lecturer about problems on the course.20 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination. Format: 4 questions from 5.

## Full syllabus

**1. Improper Integrals of the First
Kind**

Definitions of convergence and divergence for integrals of the
form ∫_{a}^{∞} f(x) dx. The Comparison Test. Absolute
convergence. Absolute convergence implies convergence. Evaluation of
∫_{0}^{∞} [sinx/x] dx. Reminder of the value of
∫_{0}^{∞} e^{− x2} dx and the deduction of the value
of
∫_{0}^{∞} e^{− (x − a/x)2} dx for a ;eq 0.
Integrals of the form ∫_{− ∞}^{∞} f(x) dx.

**2. Improper Integrals of the Second Kind**

Brief review of improper integrals of the second kind.

**3. Functions defined by integrals**

Change of order in a repeated integral of the form ∫

_{c}

^{d}∫

_{a}

^{∞}f(x,t) dt dx. Differentiation under the integral sign for integrals of the form ∫

_{a}

^{∞}f(x,t) dt.

**4. Gamma and Beta Functions**

Definition of the Gamma function and its standard properties. Definition of the Beta function. The relations

B(x,y)=Γ(x) Γ(y)/Γ(x+y) (x > 0, y > 0) and Γ(α) Γ(1−α) = π/ sin(πα) (0 < α < 1). Application to evaluation of integrals.

**5. Abscissa of Convergence.**

The existence of the abscissa of convergence c Definition of the Laplace transform.

**6. Abscissa of Absolute Convergence.**

The existence of the abscissa of absolute convergence c′. Proof that c′;eq c.

**7. Properties of Laplace Transforms**

If L(f

_{1}) and L(f

_{2}) both exist on (c,∞) and α

_{1}, α

_{2}are constants, then L( α

_{1}f

_{1}+ α

_{2}f

_{2}) = α

_{1}L(f

_{1}) + α

_{2}L(f

_{2}) on (c, ∞). If F = L(f), then F(s) → 0 as s→ ∞. The convolution f*g and the property L(f*g) = L(f) L(g). The property that F

^{(k)}(s) = (−1)

^{k}L(t

^{k}f(t)). The formulae for (i) L(f′(t)), (ii) L(f(t)/t) and (iii) L ( ∫

_{0}

^{t}f(u) du ) in terms of F = L(f).

**8. Applications**

Use of Laplace transforms to evaluate integrals. Introduction to differential equations and the use of Laplace transforms to solve ordinary differential equations and partial differential equations.

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

B |
Spiegel | Schaum outline of theory and problems of Laplace transforms | 517.35 (S) | Blackwells | Amazon |

B |
Widder | Advanced Calculus | 517 (W) | Blackwells | Amazon |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.