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[WSG25] Daily Study Group: Introduction to Laplace Transforms

Posted 4 months ago

A Wolfram U Daily Study Group, Introduction to Laplace Transforms, begins on Tuesday, February 18, 2025, at 11am CT.

Join me and a group of fellow learners to study the foundations of Laplace transforms. Our topics for the Study Group include the basic definition as an integral and its elementary properties, Laplace transforms of different types of functions, numerical approximations, inverse Laplace transforms and their properties, inverse Laplace transforms of different types of functions, complex inversion and numerical approximations for the inverse transform. A final section consists of applications to ODEs, PDEs and even fractional calculus, sums and integrals. The Study Group is intended for science, technology, engineering and math majors; teachers and professors looking for different ways of presenting Laplace transforms for their students; and anyone who wants to learn about Laplace transforms using Wolfram Language.

A basic working knowledge of Wolfram Language is recommended. If you want to get started now, try out Introduction to Differential Equations or Introduction to Complex Analysis.

Daily Study Group: Introduction to Laplace Transforms

Dates and Times: February 18 - February 28, 11am -12pm CT (4 - 5pm GMT)

Register here: https://wolfr.am/1t3TBbwv6

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POSTED BY: Juan Ortiz
19 Replies

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

POSTED BY: Mitchell Sandlin

Dear Mitch:

We used 'analytical' essentially meaning that there was a closed formula for it, particularly in the numerical evaluation lesson to distinguish between the two.

The parameter s, in the integral definition of the Laplace transform is complex, and this is the variable in the Laplace transform. That is where complex numbers come from.

POSTED BY: Juan Ortiz
Posted 4 months ago
POSTED BY: Phil Earnhardt

Thank you Phil. This is a very nice real LIVE example.

POSTED BY: Juan Ortiz
Posted 4 months ago
POSTED BY: Phil Earnhardt
POSTED BY: Carl Hahn
POSTED BY: Juan Ortiz

Thanks Juan,

But there is a ComplexPlot3D. For example:

ComplexPlot3D[(z^2 + 1)/(z^2 - 1), {z, -2 - 2 I, 2 + 2 I}]

Carl

POSTED BY: Carl Hahn

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.

POSTED BY: Juan Ortiz

I have signed up to take the quizzes for certification in the Laplace Transform DSG. I have the downloaded the textbook, the daily lessons, and the exercises (with solutions). But where are the quizzes? (I do not see any links to quizzes in the download section for the study group.)

POSTED BY: Michael Ulrey

Thanks!

POSTED BY: Michael Ulrey
Posted 4 months ago
POSTED BY: Graham Gyatt

Dear Graham:

Both seem fine to me. For problem 6, I will suggest comparing your answer with the answers provided considering that s>0 in this case. For example, you can try the command FullSimplify[yourAnswer==answerCompared, s>0] or you can cheat a bit to very using InverseLaplaceTransform on each of the posible aswers provided. For 10, make sure to set the WorkingPrecision to 10 as an option in your call to LaplaceTransform.

POSTED BY: Juan Ortiz

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?

POSTED BY: Carl Hahn

Dear Carl:

I believe that in signal analysis the Fourier transform is the way to go. Note that in the Laplace transform s is complex and in fact if h(t) is the Heaviside function and s=c+i y, then the Laplace transform of f(t) is equal to the Fourier transform of exp(-c t) h(t)f(t) depending on y. This is actually used to get the inversion formula for the Laplace transform using the Fourier transform. For more on this I can refer you to the documentation page of the FourierTransform function in the Wolfram language, where I included a subsection on signals and systems in the Applications section.

POSTED BY: Juan Ortiz
POSTED BY: James Kralik

Dear James:

WorkingPrecision->n causes all internal computations to be done to at most n-digit precision. This does not imply that the answer will be correct to that number of digits. The final results you get may have much lower precision.

Instead of comparing WorkingPrecision->10 with WorkingPrecision->20, we should compare each to infinite precision. This will require to have an analytic representation for the transform. Consider exercise 1 in the same lesson:

exact = LaplaceTransform[
   Cos[Sqrt[t^2 - 4]]/Sqrt[t^2 - 4] UnitStep[t - 2], t, s] /. 
  s -> 17/10

approximate = 
 LaplaceTransform[Cos[Sqrt[t^2 - 4]]/Sqrt[t^2 - 4] UnitStep[t - 2], t,
   1.7]

approximateWP10 = 
 LaplaceTransform[Cos[Sqrt[t^2 - 4]]/Sqrt[t^2 - 4] UnitStep[t - 2], t,
   1.7, WorkingPrecision -> 10]

approximateWP20 = 
 LaplaceTransform[Cos[Sqrt[t^2 - 4]]/Sqrt[t^2 - 4] UnitStep[t - 2], t,
   1.7, WorkingPrecision -> 20]

Now:

exact - approximate=4.50937*10^-10

exact - approximateWP10=-1.14043*10^-12

exact - approximateWP20=-3.46945*10^-18

POSTED BY: Juan Ortiz

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!

POSTED BY: Devendra Kapadia
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