Laplace Transformation Flashcards
(13 cards)
How do you calculate the laplace transformation of a function f(t)?
The integral from 0 to infinity of exp(-st)f(t)dt
Laplace transformation of 1 and exp(at)?
L{1} = 1/s
L{exp(at)} = 1/(s-a)
What is the gamma function?
Integral from 0 to infinity of exp(-t)* t^{x-1} dt
Important properties of gamma function?
Gamma(1) = 1
Gamma(n+1) = n!
Laplace transformation of t^{n}?
n!/s^{n+1}
Is the laplace transformation a linear operator?
Yes, it is.
What’s a function of exponential order?
It’s a function f(t) such that:
|f(t)| <= M*exp(Ct)
For large t, some M and C.
What can be said about the limit when t goes to infinity of f(t)/exp(Ct) when f is of exponential order?
That such limit must be less than or equal to M. Thus, it exists.
Relationship between functions of exponential order and Laplace transformation?
If a function is of exponential order, then it has a Laplace transformation.
The inverse of a Laplace transformation is unique or not?
It is unique.
Laplace transformation of f’(t)?
L{f’(t)} = sL{f(t)} - f(0)
Laplace transformation of sin(at)?
a divided by (s^2 + a^2)
Laplace transformation of cos(at)?
s divided by (s^2 + a^2)
