Laplace Transform Examples 

Isn’t that what I did?

@@drpeyam the idea is the exact same but you can keep everything on one line so the calculation is faster but perhaps less pedagogical. Great video as always.

Thanks so much!! 😊

Microsoft whiteboard :)

Thanks

But that’s too complicated

@@drpeyam Too complicated ? We integrate L(cos(t)) by parts once to get L(cos(t)) = sL(sin(t)) We integrate L(sin(t)) by parts once to get L(sin(t)) = 1 - sL(cos(t)) and we get two equations for both L(cos(t)) and L(sin(t)) Maybe it is slower but not more complicated and does not involve complex numbers

@@holyshit922 I think he's just joking. There's many paths to the same truth. Some may prefer IBP, others may prefer the complex number trick, seeing sines and cosines ultimately as exponentials with imaginary exponents.

One practical application where they are commonly used, is control system theory. A controller and the dynamics of the system it controls, are modeled with transfer functions, each of which is a Laplace domain representation of the actions it performs. Actions such as differentiating, integrating, using the original input, scaling, and linear combinations of the above. This generates each transfer function as a fraction that is built from two polynomials of s. In the time-domain, what each block is really doing, is convolution of the input with the dynamics of the system represented by the block. In the s-domain, this convolution becomes multiplication. It is often of interest to analyze the system, using the poles and zeros of the transfer function, and to combine multiple transfer functions to get overall transfer functions. The poles and zeros tell you details about the controller & system performance, such as stability, response time, and oscillation frequency. This gives you the tools to tune the details of the controller, in order to get the desired response from the physical plant you are trying to control.

Another practical application, is electric circuit analysis. We coin the concept of impedance as a generalized expansion of the concept of resistance. A resistor's impedance is its resistance of course, while capacitor and inductors have impedances of 1/(s*C) and s*L respectively. This allows you to go through the same mental exercise as combining resistances in more complicated circuits, and determine the combined impedance and transfer function of the circuit in the s-domain. Or the omega-domain, if all you're interested in, is the steady state behavior, since s becomes j*omega in the steady state (Laplace transform becomes the Fourier transform). Circuits of these combinations are ultimately differential equations that relate voltage and current. Using these transforms, the differential equation solving process, reduces to algebra.

@@carultch Thank you! Interesting stuff!

It was Heaviside and Doetsch who came up with the modern Laplace transform. Laplace had a similar transform of his own, that is more like what we call the Z-transform today. Fourier had the original transform of this kind, which uses trig functions. I suppose it comes from how versatile it is to co-integrate functions (my term for integrating a function product) with exponentials and trig, that inspired these folks to come up with these transforms. Polynomials, constants, other exponentials and trig, piecewise combinations, and linear/multiplicative combinations thereof, all have the ability to be integrated as a product with either an exponential or trig function thru a simple application of IBP.

@@carultch I remember from my calculus class that the idea behind LT was to dump the function f(t) with a negative exponential in order to make the integral of the product finite.