]> 4.3.1 Example

4.3.1 Example

By applying, the formulae [22, 858.652, 861.16, 861.01] we can represent the Laplace transform of the function f(t) = sin t t (f L2(0,)) as

f̂(x + jy) =0sin t cos yt t extdt j0 sin t sin yt t extdt = = 1 2 arctan 1 + y x + arctan 1 y x + j 4 ln x2 + (y 1)2 x2 + (y + 1)2 ,x > 0,y .

Now, (4.12), (4.13), (4.14), (4.16) yield the following results:

The above example shows that the numerical algorithms of global optimization should be applied to solve the problem (4.12).

Exercise 4.3.1. Repeat the above calculations for the function f(t) = 1l(t) 1l(t 1) L2(0,). Representing the Laplace transform of f as

f̂(x + jy) = x(1 ex cos y) + yex sin y x2 + y2 + jxex sin y y(1 ex cos y) x2 + y2

and applying formulae (4.12), (4.13), (4.14) we get: