Hi;

In trying to get a better understanding of complex numbers in the Laplace Transform, I have formed the following understanding from the webinar. Please correct me if my understanding is incorrect.

It seems that there is quite a bit of terminology associated with the complex numbers that were created during the Laplace Transform, with the first being “analytical”. My understanding of the meaning of analytical is that the derivative can be taken of the function. In other words, the function has no features which would inhibit taking a derivative, such as: cusps, jumps, esc. (just a smooth continuous function). Now sometimes in complex numbers, you encounter features such as poles (where the function runs off to infinity) or zeros (where you have a divide by zero – undefined), which these features make the function non-analytic. However, with these non-analytic functions, you can incorporate techniques such a pruning (which is piecewise definitions) to eliminate the areas of the function that are non-analytic (poles & zeros), thus making the function differentiable. If this is not correct, please tell me what I am misinterpreting.

Additionally, what causes the complex numbers of the Laplace Transform?

Thanks,

Mitch Sandlin

Today's discussion of damping reminded me of a recent find: a demonstration of damping is literally available at our fingertips! Our arms have pronators: responsible for turning our hand inwards and supinators: responsible for turning our hand outwards. The lines of pronation and supination are labeled the Deep Front Arm Line (DFAL) and Deep Back Arm Line (DBAL) respectively; they are described in the science text Anatomy Trains. A brief overview of the Anatomy Trains are available in this document; the arm lines are shown on page 16. If you co-activate the DFAL and DBAL, you will have some torsional rigidity in your hands. If you then torque one hand a bit further and rapidly release it, you can see damped torsional oscillations. See the attached video: the motion is quite similar to snapping a finger. Here's the starting position: You can see the motion in this 10s YouTube video. This may be a bit tricky to master. For the record, I can only do this with one hand; I can't "snap" my right arm this way. YMMV.

The behavior is slightly underdamped; the Quality Factor is somewhere around 2. Two interesting things: you can vary the amount of co-activation of pronation/supination; you'll see I vary it a bit in the video. Second, the damping behavior is completely orthogonal to the flexion/extension of the elbow and the flexion/extension of the shoulder. This allows for tremendous variability of postures and movements.

Co-activations can be selectively applied in all phases of cyclical movement: just-in-time tension. Our CNS knows how to roll with it all. We imitate movement by observing; we optimize movement by sensing (i.e., propriocepting) the concert of tensions inside of our body. We're constantly adjusting and tuning the movements. Learning how the body moves gives you access to move the body in new ways; it is a brilliant design.

Damping behavior is cool. It's wonderful to see the behavior modeled through the Laplace transform.

I'm a bit late to this DSG; I joined yesterday. Last night, I viewed the first session from the BigMarker video sessions. The "Download Materials" link is NOT visible in the BigMarker archive videos -- but the other links that @Cassidy inserted in the moderator's chat are visible. This appears to be a limitation of the BigMarker archive/repaly mechanism and has nothing to do with Wolfram Research. I was able to get the amoeba.wolfram.com URL by pausing Juan's presentation and taking a picture of the address bar in his browser on my iPad and parsing the URL there like an animal. ;)

I suggest the Wolfram U team be mindful of the limitations of Big Marker's video playback system and put all pertinent URLs for a course as moderator chat messages. That may mean putting whatever URL is put in the "download" area a second time in that chat window. That will help truant students get access to all the course materials on replay of the classes. Thank you, Cassidy.

I'm a bit overwhelmed with the math right now, but I appreciate where it is used in the world. That's a start. Reviewing the conversations here has been quite helpful. Thank you.

Dear Carl:

Yes, I did reply a short answer. Hrachya Khachatryan, the author of the book used ComplexPlot and it was a bit difficult for me to understand it at first too. But note the following discrepancies about the two commands. ComplexPlot[F[s],{s,2}] considers s in the rectangle with diagonal -2-2i to 2+2i in the complex plane and it shows the argument of the value of F[s] there. The key here is that F[s] is complex. Plot3D[f[x,y],{x,=2,2},{y,-2,2}] shows the value of the real function f[x,y] over the same rectangle considered with cartesian coordinates. Thus to get something that gives you similar information you will need Plot3D[Abs[F[s]]/.s->x+I y, {x,=2,2},{y,-2,2}]. Do it for s/(s^2+1) and compare.

Sorry Carl, I understood you question wrongly. Yes, is more a matter of taste but since as for now we are more interested in looking for places with zeros and poles, once you get comfortable with ComplexPlot, it is more direct, in my opinion anyways. Both work for our purposes. Thanks for clarifying.

Dear Michael:

The quizzes are in the course framework: https://www.wolframcloud.com/obj/online-courses/introduction-to-laplace-transforms/quiz-1.html

Without giving too much away, I'm wondering if there might be a typo in problem 6 and 10 of Quiz 1. For #6 I get a slightly different answer from one of them, but 'None of the Above' is not given as correct. For #10, I also get a slightly different answer from one of them, but not an exact match. Has anyone else given the Quiz a try?

Is there a formal explanation of when one should be using Laplace Transforms and when one should be using Fourier Transforms? It seems that when doing signal analysis one uses Fourier Analysis and when analyzing physical systems one uses Laplace Transforms. But then when you want to run a signal through a system you start describing the system using Fourier Transforms. But I've never read or heard someone say what exactly are the formal guidelines. When and why? Can you provide some insight into that question?

I noticed in the exercises for Part 6 that when WorkingPrecision (WP) was increased, the resulting numerical answer differed before the last digit. For example, in exercise 2, when WP->10, the result was 0.1979119699 and when WP was increased to 20, the answer was 0.19791196966502245964 yielding a difference of 3x10^10. Granted, not much of a difference, but I expected a difference of no greater than 1x10^-10. I suspect the difference is due to the accumulation of rounding "errors". There is discussion in the WP documentation that final results from internal calculations done to n-digit precision may have much lower precision. So I wonder if there are examples where a long series of calculations resulted in significant differences. If so, it would be useful to know what to watch out for. I need to review the Precision & Accuracy Control Guide in the WL documentation to see if any guidelines are suggested.

Juan is a highly-experienced instructor and has worked hard to create this wonderful introduction to Laplace transforms and their many applications.

I strongly recommend this study group to everyone!