Intro to the Laplace Transform & Three Examples
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- čas přidán 15. 06. 2024
- Welcome to a new series on the Laplace Transform. This remarkable tool in mathematics will let us convert differential equations to algebraic equations we can solve, and then convert back. In this first video we will define the Laplace Transform as an improper integral. We will see three examples: exponential functions, the step function aka Heaviside function, and the Laplace Transform of polynomials. The latter examples will make use of something called the Gamma Function and we will see it has nice properties related to factorials.
►Laplace Transforms (and more ODE topics) Playlist: • Intro to the Laplace T...
0:00 Laplace Transforms Help Solve Differential Equations
1:37 Definition of the Laplace Transform
2:46 Laplace Transform of Exponentials
5:21 Laplace Transform of Step Functions
7:21 Properties of the Gamma Function
10:31 Laplace Transform of the Gamma Function
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The wind is the soul leaving my body as i learn Laplace Transformations
😂😂
I was thinking the same thing 😂😂😂
The juxtaposition of the howls and the seriousness of the exposé is absolutely hilarious - you can't make that up. All in a sudden, I want to re-read Ginsberg poetry.
هههههههههه
It's the z transform next, and then you will have the joy of discrete signal processing! I envy you, I just loved that so much. Just think for a minute, you have all of these new vistas opening up for you to explore.
If it pains you, then you are on the wrong course.
You can use a Fourier transform (special case of Laplace transform) to filter out the wind noise in the video.
This deserves an award! LOL
It's a varying frequency, I don't think so
lmao:D I didn't think of this so I just got a new office and a new mic instead:D
@@DrTrefor Ha. Nonetheless, thanks for the great videos!
@@DrTrefor lol, we should be able to practically implement what we have learnt
some say you can even hear the screams of the horrified students...
I really heard some sound oooooooooooooooooooooohhhhhhhhhhhh
0:42. Bruh😂😂😂
Waahhhhhhgggg
Lol😂
Once it hit me - this prof looks and sounds just like my barber - the subject got a lot easier.
Shout out to Pierre-Simon Laplace for this life hack
You are one of the few that made a proper series of the Laplace transform. Much appreciated. Keep up the good work!
The Wikipedia article on this topic freaked me out. It is so outstandingly presented and I like his style.
I was just starting my journey on laplace today and i love that you uploaded this today. Honestly on of the best yt channels there is. Keep doing great things sir because you make a great impact
@@DrTrefor Thats great to hear, its sad honestly about this outbreak however it is really awe inspiring how all of us are coming together for this. Keep up the great work sir, you have helped me and many others and you will help more.
How did it go? Where are you now in terms of math?
@@brandonmohammed9092 I'd like to know too.
Cheers for these vids im currently doing Laplace Transforms for Maths Undergrad so this came at a perfect time.
Outstanding presentation! Incredible clarity. I never knew that the gamma function is the connection to the factorial, thank you so much for making this!
I remember solving these problems in undergrad!! Well explained
Happy Teaching!! ✌️✌️✅
This is an excellent, little lecture. Thank you Sir, for this and other fine series in the field of mathematics!
You're very welcome!
Love your content and I am doing my dissertation on the theory and applications of Laplace, this is a great help!
Glad it was helpful!
Thank you SO much for creating this playlist. Would be greatly appreicated if you could kindly create a PDE playlist. Your videos provide an initution approach which are incredible.
I do plan to do more pde/Fourier stuff in the future:)
The hum in the background adds a vast loneliness atmosphere. I've got different emotions while listening this lecture and lost in deep thoughts.
Professor, Your Affection with us greatful !
What are you doing step function?!
It helps illustrate concepts since its values are 1 and 0 (it's also causal).
@@paschikshehu7988 bruh
it's helping you out since you're stuck
All I can say is thank you very much, I love the way you explain.
Thank you for making this series. I was waiting for for from a long time. Thanks alot ❤
It's a GREAT HELP. Thank you again.
Thank you for your effort on this video. You should start with the Fourier Transform. Even better is to start at the Taylor/McLauren Series. Can’t expect newbies to relate to this in depth material.
Thank you for this great explanation!
Kudos Trefor - great contribution to subject - much appreciated
My pleasure!
Great explanation, this all makes so much more sense now.
Laplace transform is very important when you try to design a dynamic system.
But , what is Dynamic System?
@@im_cpk a system you're designing or modelling in which parameters change over time. For instance, in chemical engineering you use laplace transforms to design reactors and model their reactions so you know how big the reactor should be, what the reaction conditions are etc.
Fantastic presentation! Outstanding explanation with excellent examples. 💯💯💯💯💯💯💯
Thank you so much!
You're the best Sir. The explanation is very clear, much appreciated
That's really helpful and will be to everyone watching this pls continue posting vid like thse
The video becomes more exciting because he is happy to explain the topic.
Thank you for teaching!
It has been ages since i learnt and later forgot about this topic. I am now looking forward to re-learn it from you. Please speak slowly throughout so that it becomes easy to understand your words. Except for this, you are simply wonderful. Can you give examples of application of Laplace Transform in financial mathematics?
Bro just play the video on 0:75x speed ....that's good to understand us.
You Are The Best....I Can't Explain In Words...
Your enthusiasm makes your video much more interesting.
Glad to hear that!
thank you for this amazing explanation. very well presented 😌.
Glad you enjoyed it!
I like the explanation..will re listen this on repeat 🔁
Honestly.., been seeing commendable comments so far but as for me I rather feel ur not breaking this down enough and rather just jumping into solutions without even telling our it was brought about in the first place
I was thinking the same thing, then I realised our man here is being very specific about the topic he is discussing. One is expected to already have mastered primitive functions and integration. If you look at it from that angle, it makes perfect sense that the format of the video is what it is. It would make for a several hours long mammoth of a video if he had to explain this by starting from the law of identity. Besides, you only need to look at it and you should be able to tell it's mechanics, if you've done any meaningful integration in the past. After that, all you need is to cobble together a few lines of code and never have to touch this ever again.
I just say ....outstanding❤❤
Nice Explanation Thank you
your videos helped me a lot! thank you so much
clear explanation, thanks
I was doing a video on this topic. I referred to this just for additional knowledge 😊
your presentation is awesome
your way to explain this topic is so good.
Thanks a lot 😊
Awesome video, thank you
Excellent video sir.
Great Video. Thank you.
Your cadence (the way you speak) is very helpful in retaining attention and making the material easier to stick with and follow. Thank you for the video!
you are a good man, thank you
MASTER CLASS!
Great video, easy explanation ❤
Glad you think so!
Sir, can you put a video for Gamma of half integers input and how really this gamma function was brought into this form .... you really explain very well
Hmm, interesting. Utilizing e^x's property to stay the same despite being integrated, such that you can integrate over and over again? Makes a lot of sense. Question to self: what other functions do that? The sine functions do something similar, which I guess allows us to display waves over and over again.
Sine functions are linear combinations of exponential functions, so no surprise there. If you have some polynomial of the derivative D, say p(D), and you have the equation p(D) = 0, then the solutions are going to be some linear combination of exponential functions. This is because the exponential functions are the eigenfunctions of the derivative operator.
Knew I'd love it before I even watched.
This was amazing
Please make a similar playlist on the Fourier series and Transform.
It's coming actually! About 3-4 months away. Finishing Vector Calculus first then moving to differential equations and it will be part of that playlist.
Hi dr.Trevor , s is a complex number in general. And the complex numbers are not ordered set. Threrfore we can't say sa 4:03
What he means more accurately, is that the real component of s has to be greater than a, for there to exist a Laplace transform of an exponential function, e^(a*t), in order for the improper integral to converge.
Thank you for this!!!
You're so welcome!
these Videos are so great helping me for masters# student of University of Windsor ontario
The negative sign ( e raised to negative st) in the formula for laplace transform means exponential decay right? If not why else is e particularly raised to a negative power ?
thank you so much!
Danke you! Exellente explanation!
not know whcih language this hehe
How am I even supposed to understand something that's not fully explained (anywhere), like no one bothers to explain what even the purpose of laplace transforms is, you're just supposed to do it. Yet that's what I'm graded for and even if I get a good grade I would still have no clue what I'm actually doing. Kind of bizarre.
thank you so much
You're acually goated. Thnx alot
I am so damn mad that no one ever explained the Gamma Function and n! like that! I had to learn that on my own when I was in college (My Calc II professor was horrible). It was a good thing I did because when I took Differential Equations (Last semester in college), I had this insight and things were not confusing for me.
I appreciate that you explained the Gamma Function with rich substance because many students do not get the explanation to why it is equal to the factorial.
6:31 I don't understand where the "1" comes from.
This is the part where I'm supposed to input "e^{-st} * f(t) dt" where f(t) = u(t-a), am I correct?
How does f(t) become 1?
What software do you use to do this equations's animation ? Thanks
It’s all just powerpoint;)
You are great sir
I read the comment and was wondering, what wind? And while going through the video, I laughed out loud! Haha good laugh!
That comment halfway through about the howling wind made me laugh out loud. Thought it was just me going mad 😂😂
Great job
When would I use a Laplace transform? Is it for when you cannot (easily) use 'normal' integration?
Chances are, if you can't use normal methods of integration, you probably can't take the Laplace transform in the first place.
It's value comes from differential equations, and particularly differential equations involving discontinuous functions like the unit step and unit impulse. It's common that you get a diffEQ in the form of y"(t) + b*y(t) + k*y(t) = f(t), where y(t) is the function we are solving for, b and k are constants, and f(t) is a given function of t. You can think of it like a mass on a spring with damping friction, being driven to oscillate by a forcing function f(t).
When f(t) is a function like sine or cosine, earlier methods of differential equation solving work, like the method of undetermined coefficients and the second order homogeneous solution via the prototype exponential. But when f(t) is an exotic function like a piecewise function with unit steps, the Laplace transform has a great advantage.
An application where you see this, is control systems engineering.
Q. When you convert the DE to an algebraic equation why you have -2s+3?
You get this after simplifying after plugging in the initial conditions.
Hello, can I get some sources for the topic of solving differential spring equation using Laplace transform Thank you
You get a diffEQ in the form of:
M*y"(t) + D*y'(t) + K*y(t) = f(t)
where y(t) is the position function, f(t) is the forcing function, and the constants M, D, and K are the mass, damping constant, and spring constant respectively.
Then you take the Laplace transform of all terms.
M*(Y(s)*s^2 - s*f(0) - f'(0)) + D*(Y(s)*s - f(0)) + K*Y(s) = F(s)
where capital functions denote Laplace transforms
Y(s) = £{y(t)}, and F(s) = £{f(t)}
Then, you group all your terms with Y(s) to the left side, and all terms without it, to the right.
Y(s)*(M*s^2 + D*s + K) = F(s) + M*(s*f(0) - f'(0)) + D*f(0)
Make Y(s) the subject. Now you have all the inputs (forcing function and initial conditions) in the numerator, and the properties of the mass/spring/damper system in the denominator.
Y(s) = (F(s) + M*(s*f(0) - f'(0)) + D*f(0))/(M*s^2 + D*s + K)
Then take the inverse Laplace transform to get y(t). Partial fractions are very common in these inverse Laplace transforms, as you'll need to break up the big fraction into a linear combination of equations resembling standard Laplace transforms.
Laplace Transform converges (gives finite value) in ROC. How is this information (the finite value of LT) help us anyhow?
great video 👍
Why does this video has 85 dislikes? It's so helpful
ah yes, back in the day when we could all see the number of dislikes
@@nathangmail-user8860 There's an extension which has all the historic dislikes from before December 2021 and any new dislikes after are estimated from the current users with the extension, I'd recommend it 👍
Well, it gives people an opportunity to engage in the discussion and that in turn enables the algorithm to realise what a great video this is.
Otherwise you have to wonder at people even clicking on a maths video when they obviously don't like maths.
fun fact: gamma of a integer is that integer factorial-1 ! that's how people define (1/2)! even that recursion is true for non integers how cool
using the same gamma function you can even do it for complex numbers!
mind_blown.png
Ah, yes, beginning yet another one of your series. Amen.
haha you are crushing these, did you make it all the way through vector calc?
@@DrTrefor Yes sir. I feel like I have most of the intuition down, now I just need to amass a large amount of solving problems. Probably work my way through a few previous exams, that should do the trick
Sank you so much!
What is bigger, n! or infinite?🤓 Thanks for this great video.
Would like graphs instead of formal notation so geometric interpretation can intuitively explain
Thanks for the video. I need to understand how an exponent can be complex, s = σ + jω, and what it means. This is not explained. Also, as far as I know, Laplace transform is used to cenvert a continuous function in the time domain, into a function in the frequency domain. Normally, poles and zeros are presented in the complex s plane.
To understand what it means for an exponent to be complex, it all comes down to Euler's formula, to make sense of the imaginary part of the exponent. Essentially, it rotates the number in the complex plane, instead of scales it, like a real exponent does.
Given a general complex exponent of a+b*i on Euler's number, we can split the exponent with properties of exponents. a and b are real, and combine as discussed to form a complex number.
e^(a + b*i) = e^a * e^(b*i)
e^a is a positive real number, so it's just a scaling factor.
e^(b*i) is what we unpack with Euler's formula, which gives us cos(b) + i*sin(b)
What's behind Euler's formula, is the Taylor series. Use the Taylor series of e^x, and plug in an imaginary value for i*theta for x. We can do this with first principles of complex numbers, because a Taylor series is just arithmetic and integer powers. You'll get an infinite series of real terms with even exponents, and an infinite series of imaginary terms with odd exponents. These two series, are Taylor series of cosine and sine respectively.
بسیار عالی بود...احسنت...
excellent Math
I really need to go back to calculus. Any foundational books you can recommend? I have the KA Stroud book.
It's the best.
Or Rajput.
I've been struggling for weeks now with the Laplace transform method for the solution to : integral 0 to infinity of [exp (-ax2-(b/x2)) dx]. Pls help.
Excellent
Great explenation but i have got a question: if s is a variabel how can we then integrate with redpect to x? You can't integreat a function with two variabals with tespect to only one of them.
You can! Basically what you do is hold s as a constant and integrate with respect to x where you treat anything with s identically to how you would if it was a constant.
Thanks! I thought this was only possible with partial derivatives.(btw. sorry for the bad spelling I am from germany and I am only 14.
@@DrTrefor sorry that i have to ask u again but if we can treat s as a constant when integrating with respect to t coudn't we solve any differential equation like that (at least 1st order odes) . What i mean is coudn't we just multiply both sides by dx and then integrate the one side with respect to x and the other with respect to y even if the x and y terms are not sapetated?
@@erikawimmer7908hallo. Wie gut sind sie in Maths? Und was studiert sie?
i didn't even notice the wind noises until you pointed it out
love the explanation. what a cute and happy teacher
Thanx! 😊
Hi i am so confused about where e^-st e^at dt came from?? Is it a formula ?? What is it called . Thankyou its very hard to learn during this pandemic :(
Integrating a function multiplied with e^(-s*t) is the definition of the Laplace transform. The e^(a*t) is simply an example of one such function for us to take its transform.
The short answer for what s means, is complex frequency. It is a complex number, s = sigma+ j*omega, where sigma is the transient response, and omega is the steady state frequency. The j is the imaginary unit.
I'm in 10th grade like it... India
The laplace transform can be applied to both linear and non linear differential equation? true or false?
True!
With all do respect.. you had to focus on just Laplace transform and stick to it giving more examples about it. The Gama transform is another subject that confused me much while I am trying to understand Laplace, also the u function is confusing. Anyway.. your explanation is great. The winds give more horrifying feeling of the complex stuff. You could record your voice separatly and add it later to the video.
The unit step function is actually undefined at t=0
how you solved the differential equation that you showed first
I'm confused wouldn't gamma of 1 be -1? You evaluate e^(-t) at infinity, which is 0, minus e^(-t) at 0, which is 1, that would be 0-1, or negative one, so what do I not see? Help plsssss
@@CryToTheAngelz oh... you right, thanks for the help
Wow just in time
i was like, "okay interesting choice to play owl noises in the background of a math video" XD
yer a beast!