3 Properties of Laplace Transforms: Linearity, Existence, and Inverses

The Laplace Transform has several nice properties that we describe in this video:
1) Linearity. The Laplace Transform of a linear combination is a linear combination of Laplace Transforms. This will be very useful when applied to linear differential equations
2) Existence. When functions are reasonably nice - something we will call of exponential order - then the Laplace Transform converges.
3) Inverses. For L(f(t)=F(s), there is a unique f(t) for any given F(s). What this means is that we can define an inverse Laplace Transform.
This is the second video on Laplace Transforms in my Differential Equations Playlist: • Laplace Transforms and...
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[@shemsnow3711]

Man I would be so screwed without these videos. Thanks Dr Bazett!

[@osebrainquestfoundation9631]

I have found the simplest way to solve inverse of laplace transform. Thank you for the impact

[@1-10-1-4]

bounced here from the calculus III playlist. very beautifully taught, but was expecting some visualization and real-life examples same way you did there. still amazing nonetheless ♥ thank you a lot

[@africatothetop3366]

I really loves his explanations, who else was deeply focused to the extent of hearing the ambulance voice @ minute 3:49 - 3:53😂😂😂

[@saadalabbad5484]

Great channel, definitely deserves subscribing

[@DrTrefor]

Thank you!!

[@prideolea]

I love you broski. Thank you for these videos.

[@MohamedOmar-z1s]

I need you to inteoduce the proof of the inverse existence in the videos to come and big thanks for your effort!

[@continnum_radhe-radhe]

Thank you so much sir 🙏🔥🙏

[@doomslayer4276]

Thanks! Short and precise

[@Rihanna8K]

I love this channel 😍

[@austinfritzke9305]

@3:50 do you mean "continuous and of* exponential order"? @4:28 I didn't quote follow that step. @5:25 the slide was a bit quick had to rewind and pause. Still a great video. Thank you.

[@carultch]

Continuous means there are no sudden jumps, or gaps, or vertical asymptotes, or other problem points. "Of exponential order" means that the function grows no faster than an exponential function, such that multiplying by an exponential decay function makes the integral converge to a finite (i.e. non-infinite) value. For instance, tan(t) is not of exponential order, whereas polynomials are, no matter how high the degree of the polynomial.

[@stefano.a]

There is a problem in existence (minute 3:25): "s" is a complex number so it can not be greater than another number because the Complex set is not sortable

[@DrTrefor]

In the development of this series, I've been working exclusively over R. Yes, this can be extended to the complex numbers and these can be modified accordingly.

[@stefano.a]

@@DrTrefor thanks for the reply. It would be very useful a video that explain these concepts in the complex plane; In my opinion it is the most confusing part about Laplace Transform

[@brandonmohammed9092]

So for the inverse of a laplace transform, I am guessing it is not as simple as doing the lapalce transform. How would one go and compute the inverse without using the tables?

[@samridhyadav4794]

Really Helpful...Thank you so much!!

[@subaruyagami2327]

Thanks, it was really helpful!

[@DrTrefor]

Glad it helped!

[@rounakdas804]

Sir, What is your research area??

[@j.o.5957]

Linearity, existence and inverses. Seems like something linear algebra would love getting its hands on. Question to self: how do we calculate the inverse Laplace? I'm assuming I'll figure it out soon enough.

[@spyrosmanolidis8516]

2 years too late probably, but en.wikipedia.org/wiki/Inverse_Laplace_transform

[@ahmedghallab5342]

Thank you very much ❤️❤️

[@knowledge90s93]

Which of the following sequences could represent the impulse response of a stable discrete-time system? k^2 (-0.65)^k 2^k ksin(k)

[@Norm7264]

This is very clear and well presented, Trefor. It would be even better if you could improve the sound. Room acoustics are not good.

[@ermyastanru3335]

Thanks D.R

[@tramquangpho]

I have a question why plug in big number to t and the limit is M/(c -s) at the existence part

[@user-uw8rn9pc5m]

Can you make it clearer with a graph?

[@virgenalosveinte5915]

amazing videos

[@usamabinmuzaffar692]

Not to be rude... But how many denoisers did you put on the audio in premiere? lol

[@robertviragh6527]

very interesting overview, for the unique answer on getting inverses, for practical inversions how precisely do you have to know the result of the laplace transforms to get to the nearest integer of the original values say? (e.g. for normal floating point numbers can nearest integers where the input domain is -100 to 100, can they be recovered based on the floating point convluted result)? what's the algorithm for deconvoluting? Thank you for your informative and interesting video.

[@simphiwemalinga-mz8vg]

So is there a one way method to find the the unique solution of f(t) such that L{f(t)} = F(s)?

[@carultch]

The most efficient way to do it in practice, is to use a reference library of standard Laplace transforms, and match the given Laplace transform expression to a linear combination of standard Laplace transforms. There is an integral that does it more directly, but it is complicated. As an example of how to invert a Laplace transform, consider: (s - 1)/[s^3*(s + 1)*(s^2 + 1)] Construct a partial fraction expansion: A/(s + 1) + B/s^3 + C/s^2 + D/s + (E*s + F)/(s^2 + 1) Skipping ahead to the solution: 1/(s + 1) - 1/s^3 + 2/s^2 - 1/s - 1/(s^2 + 1) Known Laplace transforms that are relevant to this example: L{e^(a*t)} = 1/(s + a), which means the first term is e^(-t) L{t^n} = n!/s^(n + 1), which means the 2nd terms are polynomials of t, from a constant up to t^2 L{sin(b*t)} = b/(s^2 + b^2), which means the final term is -sin(t) 1/s^3 needs a 2 upstairs, since L{t^2} = 2!/t^2. Multiply by 2/2 to get: 1/2*(2/s^3). Its inverse is 1/2*t^2. 1/s^2 is good to go, as t 1/s is also good to go, as a constant of 1. Result: e^(-t) - 1/2*t^2 + t - 1 - sin(t)

[@freemind.d2714]

Maybe you could apply some Fourier transform to filter out that wind : )

[@DrTrefor]

well played

[@AmitVermais]

Hi ! Sir , please defined with graph and also discussed picewise continuous.

[@nyilynnseck]

Impressive presentation ! Could you show the way how you write and add these equations in videos too please?

[@shabbysing9175]

Make video on Function of exponential order in laplace transform

[@nickthepostpunk5766]

Just a quick question about existence: for existence the video states the condition s > c, but I thought s was in general complex (not real) in a Laplace transform and so I'm a little puzzled about how the (possibly) complex number s can be greater than another (possibly complex?) number c?? Thank you

[@carultch]

In this context, he's referring to the real component of s, being greater than c. The imaginary component of s, is out of the picture.

[@austinfritzke9305]

I can't really read the red font. Why not use blue? But still, great videos. The khan academy videos were pissing me off this is much better. Thank you.

[@kianushmaleki]

Here is my question: k and x are two vectors and n is the dimension then F(k) = \int e^{- k \cdot x} f(x) d^n x The dot product is defined by a metric i.e. k \cdot x = g_{ab} k^a x^b where Einstein sum convention is used. I know that the Fourier transform is not valid for a general metric. Fourier transform is certainly valid in Euclidean space. Why is the Fourier transform not valid for a generic metric? 🙃

[@minuklee6735]

Awesome!!

[@chanakyasinha8046]

Inverse laplace is weird, if integral of f(t) is F(s) then shouldn't have differentiation of F(s) be f(t) 😳 though it is definite... I don't get it 😔

[@ArthurSchoppenweghauer]

Is no one going to talk about the screaming child in the background?

[@travisjohnson8540]

He said in pt1 it's wind outside

[@projectiocon]

Sir i am from India🇮🇳 Sir i have a question that why inverse Laplace transform have not any prove ? Sir do we not get Laplace function after solving it's inverse? (Actually sir my English is not much good so please try to understand my question problem 🥺)

[@ArsalanKhan-i7c]

The inverse Laplace transform lets us take a function in the frequency domain (like 𝐹 ( 𝑠 ) F(s)) and convert it back to the original time-domain function (like 𝑓 ( 𝑡 ) f(t)). When we apply the inverse Laplace transform to 𝐹 ( 𝑠 ) F(s), we don't get 𝐹 ( 𝑠 ) F(s) back; instead, we recover the original function 𝑓 ( 𝑡 ) f(t) before the Laplace transform was applied. It might seem like there's no "proof" because we often use tables or shortcuts instead of directly calculating it, but there's actually a formal mathematical definition involving complex integration. In simple terms, think of it as a two-way translation: the Laplace transform moves from time to frequency, and the inverse brings it back from frequency to time!

[@AmitVermais]

Sir also make video on real analysis, complex analysis and modern algebra please sir please. And please comment on my message.

[@larshaji6117]

thanks a lot but there is a strange background sound distracting me

[@osebrainquestfoundation9631]

Knowledge

[@rodericksibelius8472]

I wish you can give real world examples real world application using Electronic circuit and go trough all the detailed steps on how Laplace Transform to the time domain, and vice versa. Frequency domain to the Time domain,

[@SHAHHUSSAIN]

👌

[@aniketpattnaik5757]

yo look like kevin petersen