# intuition about definition of laplace transform

by learner07
Tags: definition, intuition, laplace, transform
 P: 3,830 Laplace Transform is one of the simpler ones that has a calculus representation already. In real world, quite a bit of result can only be represented by numerical analysis which is some form of power series. Try to make sense out of that!!! People might be plotting out an input to output response and literally doing a curve fitting to develop an equation for it. Like Ratch gave, log is one of those, why do log? From my understanding, one reason is human ears response logarithmic to sound power, there must be a lot of other reason, but that's good enough for me!!! I remember the first lecture from the instructor of my Physical Chemistry that really stuck in my mind: Invention or discovery of a theory start with an idea or postulation. Then confirm or disprove using observation and experiment. If the postulation holds after scrutiny, then it is consider true and become a law. At the same time, mathematical formulas are developed to full fill the observation. Sometimes the postulation is proven by mathematical derivation. It is not the other way around that you have a theory or equation first. If you get stuck with Laplace, try quantum physics!!! I quit chemistry after I score the first in the class of the Physical chemistry and I was like 15 points above the second student. I worked in the chemistry dept. at the time and it's like the professor came to the stockroom window everyday answering my questions. Finally he said to me " Alan, you are not going to understand this, keep at it, when you get your PHD, you'll start to get the feel of it". This might not be the exact word to word, but it's very close. I quite chemistry after that, I just finished my degree and never even look for a job in that field. Between the snake biting it's tail and this, I quit. I did not know at the time it's like this in all science. Now I can really appreciate his first lecture and what he said.
P: 72
 Quote by Runei Learn that FUNCTIONS ARE VECTORS!!! This is an important part
Functions are vectors and therefore they can be represented in any coordinate system you want. Just like when you apply a transformation on a spatial vector to transform from xyz coordinates to polar, the Laplace transforms the function by projecting it onto a different space.

I tried to think of an analogy that would make this idea a little more tangible and I came up with a technical drawing. A technical drawing uses three projections to make a 3D object seem simpler to the eye. You could choose to represent a 3D object with an animated model that rotates in time, instead. If you did that, it would be very easy for your eye to interpret the picture and get a good impression of the object's shape, but measuring precise lengths and angles would be very hard, and you would have to have some kind of magic paper to display it. It becomes simpler to represent it with still images in three different locations on a piece of paper. In these two different representations, the viewing angle is either a function of time or location, and the relative length of a line may or may not depend on relative distance. We transform the coordinates to a equivalent but more easily understood system. We also choose to align the axis of the drawing in a way that is convenient for design and construction, the same object could be represented with a drawing from three other arbitrarily chosen orthoganal directions but that would be confusing to your eye.

I think this is similar to the situation of the laplace transform because its useful for transforming time-varying functions into inanimate pictures in which time dependence is represented by spatial coordinates. In S-space, periodicity becomes a length along one axis, and the slope of exponential decay becomes a length along the other axis, any functions of S-space that are multiplied together represents the convolution of their time-varying counterparts (which is why its useful for filter analysis), and phase angles look like... angles. Like in the technical drawing example, drawing things in different locations now represents something totally different and makes it easier to express certain ideas.

S-space is a nice place to work if your job is doing a lot of convolution, differential equations, and phase analysis. The laplace transform is like the train that you take to commute from regular space where you live to S-space where you work.

Once you've mastered the Laplace Transform, try the fractional Fourier Transform... I still don't get that one
P: 2,265
 Quote by learner07 i don't need math definitions. what i need is physical understanding/interpretations of what laplace transform actually is?

 Quote by yungman Read my last post, it is a definition. It is like I am making up my transform called Alan transform and is defined as A(f(t))= f(t)+1 It is absolutely useless, BUT it's a transform!!!! Difference is my transform don't worry anything and Laplace transform works!!! Their might not be any rhyme and reason at all.
there is a rhyme and reason. but the story or poem is long, not quite epic, but it's long. maybe as long as a semester long class in Linear System Theory (now they call it Signals and Systems or differential equations.

so 07, you understand what the concept of a "transform" is, right? maybe a good example is the logarithm. this transform turns a multiplication problem into an addition problem (this is because when you multiply to exponentials with the same base, you add their exponents). it also will turn a power problem into a multiplication problem. normally the latter (the transformed problem) is easier to deal with.

the Laplace Transform will turn a certain class of linear differential equations into polynomials. supposedly the latter is easier to deal with.

the real thing about Laplace is that it's a generalization of the Fourier Transform which itself is a generalization of Fourier series.

Fourier conceptuallizes functions or signals as being a sum of sinusoids, but because of Euler's formula, a sinusoidal function is actually an exponential function (with complex or imaginary exponent). and Laplace conceptualizes functions or signals as being a sum of exponential functions.

why are exponential (or sinusoidal) functions so important that you would want to use them as the basis for creating general functions? it's because exponential functions are eigenfunctions to these operations that we call Linear, Time-Invariant systems. LTI systems are very important and fundamental. if a sinusoid or an exponential function goes into an LTI system, what comes out is a sinusoid of the same frequency, or if an exponential goes into an LTI system, what comes out is the same exponential except it will be scaled by some constant.

so that is why we want to understand a signal or a function in terms of these exponential building block, because LTI systems will deal with each block simply instead of dealing with a signal as a whole.
P: 3,830
 Quote by rbj there is a rhyme and reason. but the story or poem is long, not quite epic, but it's long. maybe as long as a semester long class in Linear System Theory (now they call it Signals and Systems or differential equations.
I am no math expert, I just learn math to survive electronics. I just went back and look at 5 books I used to study ODE and PDE:

1) Introduction to Laplace Transform by W.D DAY
2) PDE with Fourier Series and BVP by Nakhle H. Asmar.
3) Differential Eq. by Zill and Cullen.
4) Elementary Applied PDE by Richard Haberman.
5) Elementary Differential Eq. and BVP by William E. Boyce.

NONE present where this equation came by. Yes, they all talked about linear transformation and all, talked about how they justify the usefulness. It is understand here that the use of LT to transform a differential equation into a much simpler form, that's the application side of it. The advantage is very obvious. But nothing on how the Laplace transform equation came by.

In fact on Page 479 of Asmar, the first page on the chapter of LT, the first line is:

"Should I refuse a good dinner simply because I do not understand the process of digestion?"
By OLIVER HEAVISIDE.

[Criticized for using formal mathematical manipulations, without understanding how they worked.]

That's what I was talking about the dream of a snake biting it's own tail for discovering the most important organic compound of the Benzene Ring. And Eisenstein had relativity in mind for years before he could come out with observation and math to prove it. Of cause, now they have books on the benzene rings, why it is a ring, relativity and all, they can characterize everything and all the theory, but that comes later, that's the interpretation of the original equation!!! BUT where they originally come from can be very funky!!! That, there may be no explanation other than just stroke of genius and creativity.

The books do talk about the particular usefulness for solving second order ODE with constant coef as the solution is in exponential form and is particular suitable for using LT. Maybe I have not gone deep enough, I have not seen using Eigenfunction in LT.
 PF Gold P: 3,166 My professor said that the Laplace transform transforms an ODE into an algebraic expression. Then on a test he gave us the following: http://www.physicsforums.com/showthread.php?t=654318. I got that the Laplace transform changed a second order ODE into a first order ODE and didn't proceed further since in the lecture he said that "we should get an algebraic expression" so I thought I had made some mistake. Now at home I realized that I should have proceeded further. Sigh.
P: 2,265
 Quote by yungman I am no math expert, I just learn math to survive electronics. I just went back and look at 5 books I used to study ODE and PDE: 1) Introduction to Laplace Transform by W.D DAY 2) PDE with Fourier Series and BVP by Nakhle H. Asmar. 3) Differential Eq. by Zill and Cullen. 4) Elementary Applied PDE by Richard Haberman. 5) Elementary Differential Eq. and BVP by William E. Boyce. NONE present where this equation came by. Yes, they all talked about linear transformation and all, talked about how they justify the usefulness. It is understand here that the use of LT to transform a differential equation into a much simpler form, that's the application side of it. The advantage is very obvious. But nothing on how the Laplace transform equation came by. ... The books do talk about the particular usefulness for solving second order ODE with constant coef as the solution is in exponential form and is particular suitable for using LT. Maybe I have not gone deep enough, I have not seen using Eigenfunction in LT.
well, yungman, it's after midnight now, so i am a little tired. maybe tomorrow i can go through the step-by-step (nowadays this should be in a Signals and Systems course) that starts with

1. Linear Time-Invariant systems (LTI), just the definitions.
2. show how the convolution summation (for discrete LTI) or convolution integral (continuous-time LTI) are derived directly from the definitions of LTI.
3. show how exponentials are eigenfunctions of LTI systems ($e^{\alpha t}$ goes in -> $A e^{\alpha t}$ comes out)
4. with Euler, show how sinusoids are also a sort of eigenfunction of LTI systems ($e^{j \omega t}$ goes in -> $A e^{j \omega t}$ comes out)
5. then generalize a little more from sinusoidal to periodic input (Fourier series). note that in deriving the Fourier coefficients, this is where the integral that ultimately becomes the Laplace Transform will first emerge.
6. then generalize a little more and let the period go out to infinity, so the periodic input becomes non-periodic. then look at that integral for the Fourier coefficients. it becomes the Fourier Integral.
7. that Fourier Integral looks just like the double-sided Laplace Integral except the Fourier has $j \omega$ in it where as, if you generalize further, the double-sided Laplace Integral has $s = \sigma + j \omega$ replacing $j \omega$.

so biologists and physiologists, if they drill down a little, do understand, for the most part, how digestion works. they don't simply say "it works, let's eat."
 P: 3,830 I don't think we are talking about the same thing. I guess another way of looking at this is: Is there a derivation of $$L(f(t))=\int_0^{\infty} e^{-st} f(t) dt$$ If this is derived from the existing theory, then I hope at least one of my 5 books would have mentioned where the equation comes from. In another word, the history of Laplace transform. We all know the application and the indefinite integral, the kernel etc. I am not particular familiar with Laplace transform, all I can based on is the 5 books I have. When you talk about LTI, is Laplace transform derive base on that? If yes, that's good enough for me, you prove your point. If Laplace transform fit into the LTI, that does not prove anything. You don't need to repeat all the theory to characterize Laplace transform, just the original history of Laplace transform, I think that's what the OP was asking since he is not interested in the definition and all. I assume he know enough about the application of it.
P: 3,830
 Quote by fluidistic My professor said that the Laplace transform transforms an ODE into an algebraic expression. Then on a test he gave us the following: http://www.physicsforums.com/showthread.php?t=654318. I got that the Laplace transform changed a second order ODE into a first order ODE and didn't proceed further since in the lecture he said that "we should get an algebraic expression" so I thought I had made some mistake. Now at home I realized that I should have proceeded further. Sigh.
That's the application side, that it transform

$$L(y')=-y_0+sL(y) \;\hbox { and } \; L(y'')= -y_0' -sy_0 + s^2L(y)$$

Or

$$L( \cos(at))= \frac s {s^2+a^2}$$
 P: 3,830 This is the history I found: http://en.wikipedia.org/wiki/Laplace_transform The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called z transform) in his work on probability theory. The current widespread use of the transform came about soon after World War II although it had been used in the 19th century by Abel, Lerch, Heaviside and Bromwich. The older history of similar transforms is as follows. From 1744, Leonhard Euler investigated integrals of the form $$z = \int X(x) e^{ax} dx \hbox { and } z = \int X(x) x^A dx$$ as solutions of differential equations but did not pursue the matter very far.[2] Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form $$\int X(x) e^{- a x } a^x dx$$ which some modern historians have interpreted within modern Laplace transform theory.[3][4][clarification needed] These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.[5] However, in 1785, Laplace took the critical step forward when, rather than just looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form: $$\int x^s \phi (x) dx$$ akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.[6] Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space as the solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.[7] It is a continuation of Euler and Lagrange's development. Now the question is what's is the history of Euler's.
 P: 519 Fantastic thread. I want to take some time to digest this during break. My encounter with laplace was unpleasant.
 Sci Advisor P: 3,015 indeed a neat thread. I am grateful to the pointers to early mathematicians.... My experience with Laplace transforms was in automatic controls, which would be impossible without them. i read someplace that the math of feedback systems was developed by Descarte and shelved. Of course it was shelved, for there was no automation at the time. During WW2 the Germans revived Descarte's old math for their rocketry programs and their textbooks were among the war prizes we brought back. Along with the rocket scientists to explain them. I passed my controls courses by using Laplace as a tool whose workings i did not comprehend. I'd suggest that repetition might be your quickest way to conquer this. At least it'll make you familiar with the patterns, and for me that usually leads to insight. But I never mastered Laplace's transform. old jim
P: 3,830
 Quote by jim hardy I passed my controls courses by using Laplace as a tool whose workings i did not comprehend. I'd suggest that repetition might be your quickest way to conquer this. At least it'll make you familiar with the patterns, and for me that usually leads to insight. But I never mastered Laplace's transform. old jim
Being able to understand the theory and being able to design are not necessary related. There are people that are very good in theory and there are people that can design!! To me, there is an artistic component in electronic design, I made use quite a bit of my experience as a musician when I was young.

I know people that are very strong in theory but cannot design squad!! As an engineer, it's the result that is important, people hire one that can design good electronics using fortune telling than someone that can talk theory and can't produce. AND it is more irritating to have someone that can't get the job done and then argue with you in theory why it can't be done!!! Of cause, best will be good in both, but I'll take someone that can design any time of the day.

I am not good with Laplace transform and Fourier transform. I study these for signal and system, modulation. But after I studied these, then I found out that I still need statistic and probability to really understand the books. I kind of drop it all together and choose RF, EM instead, I rather get into transmission line, RF amplifier and antenna design. Those are totally different animal all together. I spend the most time on that instead. I still use Bode Plot for closed loop feedback designs!!! Simple, but works for me.

I found the first lecture and what the professor said to me described in post #19 is so important, that really open my eyes in the field of science. There are a lot more "guessing", "opinion", "eagle" and "politics" in science than people realize. There are a lot of theories, math come after the fact..........It's like Monday morning quarterbacking, people analyzed to death why the team lost the game, what is the reason and all. Or even why a team won a game.
 PF Gold P: 368 I always have made use of laplace transforms to integrate differential equations and solve them in less hastle. In terms of what it physically means, the relationship before the laplace transform is where the real intuition would lie, but then if we take this intuition - take the infinite sum of the function from zero to infinity, the intuition about the initial problem still exists. If we take the inverse laplace, the simplifications we make in that domain are perfectly valid. I agree with yungman (not to put words in his mouth, but,) it's best described as a "reality" in terms of mathematics.
P: 2,265
 Quote by yungman I don't think we are talking about the same thing. I guess another way of looking at this is: Is there a derivation of $$L(f(t))=\int_0^{\infty} e^{-st} f(t) dt$$ If this is derived from the existing theory, then I hope at least one of my 5 books would have mentioned where the equation comes from. In another word, the history of Laplace transform. We all know the application and the indefinite integral, the kernel etc. I am not particular familiar with Laplace transform, all I can based on is the 5 books I have. When you talk about LTI, is Laplace transform derive base on that? If yes, that's good enough for me, you prove your point. If Laplace transform fit into the LTI, that does not prove anything. You don't need to repeat all the theory to characterize Laplace transform, just the original history of Laplace transform, I think that's what the OP was asking since he is not interested in the definition and all. I assume he know enough about the application of it.
the OP was asking how this laplace transform actually works?

$$\mathcal{L}\{f(t)\}=\int_0^{\infty} e^{-st} f(t) dt$$

is derived, and i am saying it is an extension of the Fourier Transform which is an extension of Fourier series and you'll see the first integral that evolves to be the F.T. and L.T. from the derived Fourier coefficients in the F.T.

and the reason why sinusoidal and exponential functions are used as the basis functions for F.T. and L.T. are because they are eigenfunctions for Linear Time-Invariant systems.

it's not magic, and the definition of L.T. did not appear by magic. there is a rhyme and reason to it and a decent modern course in Signals and Systems (what we used to call Linear System Theory) would spell this out.
 Sci Advisor P: 3,015 i'm not mathematically inclined, let alone gifted. ODE was as far as i went. Might Euler's equation be part of the intuitive explanation OP inquired about?? when we multiply some arbitrary function by e^st where s includes a real term σ and a jω and has correct dimension (t^-1), it's not difficult for me to imagine that operation multipies our function by sine/cosine and exponential functions , analagous to a frequency sweep of a circuit plus ringing it with pulses , and integrating over 0 to ∞ collects the results and makes it somehow represent what a previous poster called a transform into a new plane,,, frequency dependent behavior being its ordinate and exponential behavior its abcissa? Please excuse this musing of a math ignoramus, its just i woke up in middle of night with that question. My alleged brain chews on concepts like this for months and ususlly discards them maybe somebody can accelerate that rejection process for me, or say it's worthy of more thought. Right now it's the best lead i've run across. It ties the transform to something i can conceive of doing with hardware. Thanks guys, old jim (a chid of the lesser gods)
 P: 142 Bumping because I find it mind boggling that such a neat concept is so poorly understood. There isn't much to fuss about regarding the intuition behind the Laplace transform. You guys should check out OCW:s differential equations lecture series, prof. Mattuck gives a fantastic explanation. The Laplace transform is simply the continuous analogue of a power series expansion, where instead of summing an infinite amount of c_n x^n, with the terms separated by 1, you're integrating an infinite amount of e^x terms, each separated by dx. I remember when they first went over power series expansions in out Calc 1 class, I thought to myself "hmm, I wonder if you can use integrals instead of sums to represent functions". Turns out you can, and the result is the Laplace transform!
P: 55
 Quote by Gauss M.D. Bumping because I find it mind boggling that such a neat concept is so poorly understood. There isn't much to fuss about regarding the intuition behind the Laplace transform. You guys should check out OCW:s differential equations lecture series, prof. Mattuck gives a fantastic explanation. The Laplace transform is simply the continuous analogue of a power series expansion, where instead of summing an infinite amount of c_n x^n, with the terms separated by 1, you're integrating an infinite amount of e^x terms, each separated by dx. I remember when they first went over power series expansions in out Calc 1 class, I thought to myself "hmm, I wonder if you can use integrals instead of sums to represent functions". Turns out you can, and the result is the Laplace transform!
Speaking just from my experience in engineering school, the Laplace transform is, more or less, as a "gift from the gods". (borrowing the phrase from this nice writeup).

Understanding that $e^t$ is an eigenfunction of $d/dt$ can help understand that the Laplace transform works by design and not accident.

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