Usage of inverse Laplace transform

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:

$$f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \lim_{T\to\infty}\int_{ \gamma - i T}^{ \gamma + i T} e^{st} F(s)\,ds$$

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Are you expecting a practical usage of ILT, for example in engineering? –  Babak S. Dec 8 '12 at 11:47

To actually evaluate the inverse Laplace transform integral, we must perform an integration in the complex plane. Many times, this means using the theorems of complex analysis, notably Cauchy's residue theorem. Because many students are introduced to the Laplace transform before they take complex analysis - many times, such students won't ever take a course in complex analysis - most first time lessons focus on using tables, to your chagrin.

If you are comfortable with integration in the complex plane, however, evaluating the ILT provides a terrific exercise for your contour integration. For example, consider a LT $\hat{f}(s)$ having poles in the complex plane. Rather then try to actually evaluate the ILT integral directly, we consider the integral over the closed contour $C$:

$$\frac{1}{i 2 \pi} \oint_C ds \, \hat{f}(s) \, e^{s t}$$

What is $C$? It turns out that our choice of $C$ will depend on the sign of $t$. When $t>0$, we will need to add a circular contour to the left of the vertical line of integration of the ILT. When $t<0$, we close to the right instead. The reason for this is that we need the integral over the circular contour to vanish as the radius $T$ of the circle approaches $\infty$. To see this, parametrize the integral there as $s = R e^{i \phi}$; then the integral over a leftward-circular contour looks like

$$i T\int_{\pi/2}^{3 \pi/2} d\phi \, e^{i \phi} \, \hat{f}(T e^{i \phi}) e^{t T \cos{\phi}} e^{i t T \sin{\phi}}$$

Note that $\cos{\phi} < 0$ within the integration region; thus the integral only vanishes when a) $t > 0$, and b) $\lim_{T \to \infty} T \hat{f}(T) = 0$.

This brings us to my next point: causality. The ILT $f(t)$ must be zero when $t<0$. This means that $\gamma$ must be chosen so as to be to the right of the pole with the largest real part. That way, when the contour closes to the right, the integral over the closed contour will be zero (as it encloses no poles) and, hence, the ILT is zero by design. Note that it doesn't matter what $\gamma$ is, so long as there are no poles to the right of the line $\Re{z}=\gamma$.

In the simple cases we find from, for example, solving ordinary differential equations with constant coefficients, we will have $\hat{f}(s)$ being a rational function of $s$. In such cases, we will simply have

$$\frac{1}{i 2 \pi} \int_{\gamma-i \infty}^{\gamma+I \infty} ds \, \hat{f}(s) \, e^{s t} = \sum_k \text{Res}_{s=s_k} [\hat{f}(s) \, e^{s t}]$$

As long as you know how to evaluate these residues, then evaluating such ILTs is straightforward. Let's look at a concrete example: solve the differential equation

$$x'' + 4 x= 1$$

with $x(0) = x'(0) = 0$. We apply LTs to both sides and we get

$$\hat{x}(s) = \frac{1}{s (s^2+4)}$$

To obtain $x(t)$, we sum over the residues of $\hat{x}(s) e^{s t}$ at the poles at $s=0$, $s=2 i$, and $s=-2 i$. At $s=0$, the residue is

$$\lim_{s \to 0} s \frac{e^{s t}}{s (s^2+4)} = \frac14$$

The other residues are found similarly:

$$\lim_{s \to 2 i} (s-2 i) \frac{e^{s t}}{s (s^2+4)} = \frac{e^{i 2 t}}{-8}$$

$$\lim_{s \to -2 i} (s+2 i) \frac{e^{s t}}{s (s^2+4)} = \frac{e^{-i 2 t}}{-8}$$

The ILT is the sum of the residues, or

$$x(t) = \frac{1-\cos{2 t}}{4} = \frac12 \sin^2{t}$$

There are times, though, when you need other techniques of contour integration. For example, sometimes you may encounter a LT having a branch point in $s$. One example I have encountered here is $\hat{f}(s) = e^{-\sqrt{s}}$. Here is my evaluation of this ILT. Note how I treated the contour. Despite the fact that there are no poles and hence no residues, the multivaluedness of the LT provides the ILT I sought. I have many more worked examples here.

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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://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

http://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

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+1 Enjoy, indeed, -A! Wishing you well today! Can't wait to hear how it goes. –  amWhy May 9 '13 at 13:46
Me too (I get VERY nervous in such situations)...but once you get going, your passion will take over! –  amWhy May 9 '13 at 14:12