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MM354 Assignment 1 This is the first of two assigments which form the assessment for the second part of the module. Each will be marked out of 40,...

1. Inversion of the Laplace transform
The Laplace transform of a function f (t) of a function of exponential order is defined as
F(s) = ∫∞
0
e−st f (t) dt.
This transform had many applications in applied mathematics. It can be shown that the inverse
transform can be obtained as the Bromwich integral,
f (t) = 1
2πi ∫γ + i∞
γ − i∞
ezt F(z) dz.
The integrand here is over the line where is chosen sufficiently large that all
singularities of lie to the left of the line. In practice we evaluate the integral as the limit as
of a contour integral over the contour shown below. Note that the curved part is a
semicircle of radius centred on the point , not . We assume throughout that is
real.
Re (z) = γ γ
F(s)
R → ∞
R z = γ z = 0 t
R
−R
Re( z )
Im( z )
γ
The Bromwich contour
We are going to use this contour to evaluate the inverse transform of two functions. Both of
these can be done easily by elementary methods, but they serve to illustrate the method.
(i) Let
F(s) = 1
(s + 1)2 + 1
.
Since the poles of this are in the left half plane, we can chose γ = 0.
Use the Residue Theorem to find f (t).
Note that to obtain full marks you must give a proper proof that the integral around the curved
section of the Bromwich contour tends to zero as goes to infinity. In particular, you need to
find the maximum value of on the contour.

MM354 Assignment 1 This is the first of two assigments which form the assessment for the second part of the module. Each will be marked out of 40, but contribute 20% of the total module mark. Hand in the work to the CMIS office by 8.15am on Friday 21st May 2010. 1. Inversion of the Laplace transform The Laplace transform of a function of a function of exponential order is defined as f ( t ) F ( s ) = 0 e st f ( t ) d t . This transform had many applications in applied mathematics. It can be shown that the inverse transform can be obtained as the Bromwich integral , f ( t ) = 1 2 π i γ + i γ i e zt F ( z ) d z . The integrand here is over the line where is chosen sufficiently large that all singularities of lie to the left of the line. In practice we evaluate the integral as the limit as of a contour integral over the contour shown below. Note that the curved part is a semicircle of radius centred on the point , not . We assume throughout that is real. Re ( z ) = γ γ F ( s ) R → ∞ R z = γ z = 0 t R -R Re( ) z Im( ) z γ The Bromwich contour We are going to use this contour to evaluate the inverse transform of two functions. Both of these can be done easily by elementary methods, but they serve to illustrate the method. (i) Let F ( s ) = 1 ( s + 1 ) 2 + 1 . Since the poles of this are in the left half plane, we can chose . γ = 0 Use the Residue Theorem to find . f ( t ) Note that to obtain full marks you must give a proper proof that the integral around the curved section of the Bromwich contour tends to zero as goes to infinity. In particular, you need to find the maximum value of on the contour. R | e zt | Continued over the page Page 1 MM354 Assignment 1 Fri, Apr 30, 2010 S.W.Ellacott
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Recall e Re ( w ) . | e w | = (12 marks) (ii) Now consider the case F ( s ) = 1 s + 1 . Again the pole is in the left half plane, so we can chose . γ = 0 What problem arises when we try to show that the integral on tends at zero as goes to infinity? There is a theorem which overcomes this problem and which is found in most complex variable textbooks. Look up and quote the theorem, and use it to complete the argument correctly. Γ R (13 marks) 2. Numerical evaluation of complex integrals Numerical evaluation around smooth curves in the complex plane is usually carried out using the trapezium rule. (You will find out later why the trapezium rule is particularly good for this purpose.) For our purposes we just need integrals around the unit circle . So we require to evaluate | z | = 1 | z | = 1 f ( z ) d z . Set and show that the point composite trapezium rule gives z = e i θ N + 1 | z | = 1 f ( z ) d z = 2 π 0 f ( e i θ ) i e i θ d θ 2 π i N N 1 k = 0 f ( e i θ k ) e i θ k θ k = 2 π k N . where For the case , write an Excel spreadsheet to implement this rule. Note that Excel does not support complex arithmetic, so you will need to calculate the real and imaginary parts of the summand explicitly. N = 16 Test your spreadsheet on at two integrals evaluated exactly using the Cauchy formula or the residue theorem. For at least one of your test problems, both the real and imaginary parts of the integral should be non-zero. You should find that the trapezium rule is very accurate. You should submit printouts for both of your test problems, but you need not submit an electronic version of the spreadsheet. (15 marks) You will need this spreadsheet for the second assignment. S.W.Ellacott Fri, Apr 30, 2010 MM354 Assignment 1 Page 2
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