Laplace transform: Why do we only care about Re(s)

When encountering laplace transformations you are bound to work with certain conditions to make those functions converge. So why do we only care about the real part of this complex valued s?

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[Vhaanzeit]

Control Engineer here. There's a really nice practical connection to this as well from the Maths perspective. The Laplace Transform of Linear DEs provides a frequency domain description of the problem and when considering what practical frequencies exist for real systems it's very quickly apparent that 'negative' frequencies do not exist which ties in with the Laplace Transform being taught as a one sided integral from 0- to infty with the Bilinear transform sometimes being taught for fun. Furthermore, this ties deeply into causality. Causal systems depend only on the current and past inputs applied to it and not future inputs. If a system were to start reacting to things that it knows will happen in the future, then you have something really really whacky (non causal). Mathematically this is similar to being in the negative frequency and/or time axis.

[quantumchill]

Yikes, e to the negative complex infinity, knew that can't be good.

[MrRyanroberson]

heh. so, how about this:
all of trigonometry can be derived from four diagrams and one equation with a clever attitude.
diagram 1: proof of interior angles theorem
diagram 2: proof of area of triangle and angle sum formulas.
diagram 3: square rotated within a greater square (pythagorean theorem proof)
diagram 4: right triangle of hypotenuse (tan+cotan) legs secant, cosecant, altitude of 1, and altitude of the two sub-triangles labeled sine, cosine appropriately.
equation: e^it=cos(t)+i* sin(t)=(taylor series of cosine)+i* (taylor of sine)

it's amazing how far this goes, and how fast that it doesn't take many proofs to complete trigonometry. as far as i know, all you have to do to finish is to write out all the ratios between triangles and take the product of the equation by itself but having a different input.
cos(t+p)+i* to prove multi-angle formulae.

did you ever do the proof for that equation?

[AdamSmith-jblf]

Great video for the exam I'm in right now. PRAISE BE TO ONLINE EXAMINATIONS

[andresxj]

I thought that -s was equal to ln(x), which was going from 0 to the negative infinity. I've seen somewhere that, when we define this operation (the Laplace transform), we decide to say that ln(x) (which is what we had originally) is equal to -s in order to have the function in terms of s. Why do you say s is an element of the complex numbers?

[nitigyajoshi]

In one question, we were given f(s) and we had to find inverse laplace tranform. But with it one condition was given. Re{s} < -1. will this affect the solution at all?

[JarogniewBorkowski]

Could derive Mellin's inverse formula to the Laplace transform?
Your videos are awesome.