3) Let ƒ : (0, ∞) → C denote a bounded continuous function. The Laplace transform of f = f(x) is (LĐ)(z) = fe) == da =: F(2) for Rez > 0. The formula for the inverse Laplace transform is 1 (L~¹F)(x) = 2 i fr [F(z)e™z dz where c > 0 and Ã' is the straight line with parameterization z(y) = c+iy, -∞ 0 with Laplace transform F(z) = ½. Fix c> 0 and let Ãc,R denote the straight line with parameterization z(y)=c+iy, -R≤y≤R. Use complex variables to prove that 1 1 x>0 2(x) = = = = = = = ² ² ² = { 1 te ez dz lim 2πi R→∞, x < 0 Tc, R and determine the value of g(x) for x = 0. for for
3) Let ƒ : (0, ∞) → C denote a bounded continuous function. The Laplace transform of f = f(x) is (LĐ)(z) = fe) == da =: F(2) for Rez > 0. The formula for the inverse Laplace transform is 1 (L~¹F)(x) = 2 i fr [F(z)e™z dz where c > 0 and Ã' is the straight line with parameterization z(y) = c+iy, -∞ 0 with Laplace transform F(z) = ½. Fix c> 0 and let Ãc,R denote the straight line with parameterization z(y)=c+iy, -R≤y≤R. Use complex variables to prove that 1 1 x>0 2(x) = = = = = = = ² ² ² = { 1 te ez dz lim 2πi R→∞, x < 0 Tc, R and determine the value of g(x) for x = 0. for for
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:3) Let f (0, ∞) → C denote a bounded continuous function. The Laplace
transform of f = f(x) is
(Lƒ){(z) = √°°ƒ(a)e¯ª² da =: F(z) for Rez>0.
The formula for the inverse Laplace transform is
(L¯¹F)(x) =
1
2πί
[F(z)e=²z dz
where c> 0 and Te is the straight line with parameterization
z(y) = c+iy, -∞ <y<∞0.
Consider the function ƒ(x) = 1 for x > 0 with Laplace transform F(z) = ¹.
Fix c> 0 and let TC,R denote the straight line with parameterization
z(y) = c+iy, -R≤y≤R.
Use complex variables to prove that
1
1
g(x):
=
lim
Sron
ez
2πi Rito - e²² dz
={ 1
x > 0
x < 0
2
c,R
and determine the value of g(x) for x = 0.
for
for
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