I have few confusions while starting Laplace Transform. So far I have studied, Fourier series and Fourier Transform. The basic difference which I found from different books is Fourier Transform is only considered the imaginary part whereas the Laplace transform considers both real and imaginary for general values.
i) I want to ask that, is it only difference we have in Laplace and Fourier Transform?
Then i saw the two different equations of Laplace transform which is bilateral Laplace Transform $$ X(s) = \int _{-\infty}^{+\infty} x(t) e^{-st} dt$$
Whereas in the second equation of Laplace Transform is called unilateral Laplace Transform and it defined as:
$$ X(s) = \int _{0}^{+\infty} x(t) e^{-st}dt $$
It omit the negative part and only have for $t>0$
ii) Here I want to ask that what is the reason of omitting the $t<0$ part?
And Lastly, there was an example $$ x(t)=e^{-at}u(t)$$
After applying Laplace transform it was written The transform exists only if $Re(s+a)$ is possitive.
iii) Now here is am confuse that why it took for possitive?