The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), which is a unilateral transform defined by:

where s is a complex number frequency parameter , with real numbers and . In the case of the Laplace transform, we are trying to write f(t) NOT as a sum of sine waves (like in Fourier Transform), but as a sum of exponentials.

The Laplace transform of a function is just like the Fourier transform of the same function, except for two things. The term in the exponential of a Laplace transform is a complex number instead of just an imaginary number and the lower limit of integration doesn’t need to start at -∞. The exponential factor has the effect of forcing the signals to converge. That is why the Laplace transform can be applied to a broader class of signals than the Fourier transform, including exponentially growing signals.

I am currently working as a PhD researcher in Delft University of Technology at the Delphi consortium. I use Acoustic wavefield Imaging, specifically seismic wavefield to image the subsurface of the earth. I have been developing imaging methods that use `multiple wavefields' to push the limits of conventional imaging methods.
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