Kenneth Kuttler

764 CHAPTER 37. SOME FUNDAMENTAL FUNCTIONS AND TRANSFORMS

Is this equal to F (s)? Consider a large circular contour of radius M where M > |s| andorient it counter clockwise about s as shown in the following picture. Denote this orientedcurve as ηM .

x = c

Then from the estimate assumed on F,∣∣∣∣∫ηM

F (z)z− s

dz∣∣∣∣≤ C

Mα

1M−|s|

2πM

Now as M→ ∞, this converges to 0. Therefore, from the usual Cauchy integral formula,

F (s) =1

2πi

(∫η̂R

F (z)z− s

dz+∫

ηM

F (z)z− s

dz)

Now take a limit of both sides as M→ ∞ and you obtain

F (s) =1

2πi

∫η̂R

F (z)z− s

dz =1

2πi

∫ηR

F (z)s− z

Thus this shows the following interesting proposition. This proposition shows conditionsunder which a meromorphic function is the Laplace transform of a function which happensto be given by the Bromwich integral and they are the conditions used earlier.

Proposition 37.5.6 If Re p < γ for all p a pole of F (s) and if F (s) is meromorphic andsatisfies the growth condition 37.8, and if f (t) is defined by the Bromwich integral, thenF (s) is the Laplace transform of f (t) for large s.

37.6 Exercises1. Let F (s) = 2

(s−1)2+4so it is the Laplace transform of some f (t). Use the method of

residues to determine f (t).

2. This problem is about finding the fundamental matrix for a system of ordinary dif-ferential equations

Φ′ (t) = AΦ(t) , Φ(0) = I

having constant coefficients. Here A is an n× n matrix and I is the identity matrix.A matrix, Φ(t) satisfying the above is called a fundamental matrix for A. In thefollowing, s will be large, larger than the magnitude of all poles of (sI−A)−1.

(a) Show that L(∫ (·)

0 f (u)du)(s) = 1

s F (s) where F (s)≡L ( f )(s)

(b) Show that L (I) = 1s I where I is the identity matrix.

764 CHAPTER 37. SOME FUNDAMENTAL FUNCTIONS AND TRANSFORMSIs this equal to F (s)? Consider a large circular contour of radius M where M > |s| andorient it counter clockwise about s as shown in the following picture. Denote this orientedcurve as Ty.y7XYThen from the estimate assumed on F,F 1| @) y)< nMnw 2-9 "| = ME MIsNow as M — 9, this converges to 0. Therefore, from the usual Cauchy integral formula,F(s)= - (/ PO ary “0 «)fr =—S ny <—SNow take a limit of both sides as M — o and you obtain1 F 1 FFijaxef Peet f FO271 Jap Z—S 27 Ing S—ZThus this shows the following interesting proposition. This proposition shows conditionsunder which a meromorphic function is the Laplace transform of a function which happensto be given by the Bromwich integral and they are the conditions used earlier.Proposition 37.5.6 If Rep < y for all p a pole of F (s) and if F (s) is meromorphic andsatisfies the growth condition 37.8, and if f (t) is defined by the Bromwich integral, thenF (s) is the Laplace transform of f (t) for large s.37.6 Exercises1. Let F(s) = wsresidues to determine f (ft).so it is the Laplace transform of some f (t). Use the method of2. This problem is about finding the fundamental matrix for a system of ordinary dif-ferential equations®' (t) =A®(t), (0) =/having constant coefficients. Here A is an n x n matrix and J is the identity matrix.A matrix, @(t) satisfying the above is called a fundamental matrix for A. In thefollowing, s will be large, larger than the magnitude of all poles of (s/ —A),(a) Show that # (0? fw) du) (s) = ‘F (s) where F (s) = 2 (f)(s)(b) Show that (J) = 1] where J is the identity matrix.