At my current study level in college, use of inverse Laplace transform is not mentioned well - textbooks say "use tables." So, can anyone show me how to use inverse Lapalce transform? And also proof?
Edit: I am not asking how to use tables to solve inverse Laplace transform - I ACTUALLY want to know how to solve inverse Laplace transform using the inverse Laplace transform formula:
$$f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \lim_{T\to\infty}\int_{ \gamma - i T}^{ \gamma + i T} e^{st} F(s)\,ds$$
There are many wonderful resources for learning the Laplace Transform including sites, papers and books.
Here are some examples:
http://en.wikipedia.org/wiki/Laplace_transform
http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/ode/laplace/solve/solve.html
www.youtube.com/watch?v=Z_wQvCyKjwE
http://web.firat.edu.tr/agulucar/AM/laplaceforengineer.pdf
Here is a problem book with solutions: http://cazelais.disted.camosun.bc.ca/175/laplace-book.pdf
https://mathoverflow.net/questions/16274/fourier-vs-laplace-transforms
Book: The Laplace Transform: Theory and Applications
Book: Fourier and Laplace Transforms
Book: An Introduction to Laplace Transforms and Fourier Series
You can also compare these to other transforms such as the Fourier Transform, z-Transform, Discrete-Time Fourier Transform and Discrete Fourier Transform.
Enjoy -A