# How to learn Integral Transform?

I major in Electronic Engineering when back in college. I learned the Fourier Transform, Laplace Transform, Z Transform and wavelet Transform.

But I always feel a lack of thorough understanding of the mathematical logic behind these calculations. So when I do such calculations, it's more like follow my habit than logical reasoning.

I think this is because I don't have a complete picture of the background math knowledge. So I want to spend some time (about one year) to make up for it. Otherwise it'll be a pity for my life.

My math background:

• Calculus
• Linear Algebra

Could some one list the knowledge I should learn to fully understand Integral Transform? (And some book recommendations are appreciated.)

(I choose the related tags based on my own guess. Excuse me if it is not appropriate.)

During my searching, I found some articles/books useful to me. I will keep adding links to them below. Maybe they are just remotely related to this question. But they do make me aware of something new.

The Axiom of Choice in an Elementary Theory of Operations and Sets

• I've wanted to ask a similar question. Why is it the books almost always start here's the Laplace transform use it to solve differential equations? I'm always left puzzled as to how anyone ever came up with the idea in the first place. An analogue to power series is about the closest I 've got to understanding.
– Karl
Mar 13, 2015 at 6:26
• @Karl If you really want an answer, you should post this as a question, rather than as a comment to someone else's question. Mar 13, 2015 at 6:33
• @Karl You can star my question as your favorite. And you are welcome. ;) Mar 13, 2015 at 8:42
• @neuguy I'll pose a question about my understanding of integral transforms but I fear it'll read as a muddle of confusion and receive down votes. Meantime I'll star this.
– Karl
Mar 13, 2015 at 10:12
• @Karl A question on the motivation for the Laplace transform seems far from a muddle of confusion to me. Also, don't worry about downvotes - what do you care if an anonymous goon on the internet thinks your question is stupid? Besides, as they say, there is no such thing as a stupid question. Mar 13, 2015 at 19:08

Laplace transforms were derived in a very strange way by Oliver Heaviside, who is considered by many to be the Father of modern Electrical Engineering. He created 'operator' methods for solving ordinary differential equations. (The 'D' operator was Heaviside's notation, and the algebraic method was his, including using partial fractions and his 'cover up' method for decomposing into partial fractions.) Most of what he did was not very rigorous, but it was brilliant, it worked, and he always checked his answers. The reason you have trouble tracing back to the source is because Heaviside was so arrogant and nasty to people at the time, that they vindictively set out to keep his name out of everything. Honestly. He used to openly and viciously insult Lord Kelvin. Heaviside was banned from publishing several times throughout his life for his open attacks through Journal articles.

Heavside deliberately set out to turn differentiation into multiplication, and he came up with expressions that morphed into something similar to what is now called the Laplace transform. But it didn't start off as something called the Laplace transform; when people found integral expressions similar to what Heaviside was using that could be named after someone else, they jumped at the chance to write Heaviside's name out of it. Heaviside noticed that time evolution operators for time-invariant systems (such as circuits) would have an exponential property. That is, if the solution operator acted on a state $x$ at time $0$, then the state $S(t)x$ at a time t seconds later when evolved again by $t'$ seconds should be the same as the state obtained by evolving the original state by $t+t'$ seconds. In other words, the solution operator would satisfy $S(t')S(t)x=S(t'+t)x$. Very abstract, very general for such systems, and obviously leading to something exponential. That's where the exponential in the Laplace transform comes from, and that's the level Heaviside worked at during the late 1800's! His operator methods allowed him to solve problems nobody else at the time could; otherwise people at the time would have gladly ignored Heaviside.

We now recognize that many differential equation solution operators can be viewed in this abstract way of Heaviside. For example, if you have Laplace's equation on a half plane, $x \in\mathcal{R}$, $y > 0$, and you look at a solution operator that takes boundary data $f$ at $y=0$ to a function $g=L(y)f$ at $y > 0$, which is the slice of the solution at $y > 0$, and then solve Laplace's equation with that new boundary function, and look at the slice $L(y')g=L(y')L(y)f$ of the new solution, you should get $L(y'+y)f$. There's a general exponential property of time evolution operators; and there's a general exponential property connected with uniqueness of solutions of differential equations. The Laplace transform is intimately connected with these ideas. $C_{0}$ semigroup theory is based on this observation, and is also connected with the Laplace transform. The operator formalism is definitely traceable back to Heaviside.

Most integral transforms arise out of integral 'sums' of eigenfunctions of second order ordinary differential equations on $[0,\infty)$ or $(-\infty,\infty)$. Because the integrals use eigenfunctions, these 'transforms' turn the original operator into multiplication by the eigenvalue parameter. For example, the Fourier transform originated in trying to write a function $f$ as an integral sum of eigenfunctions of $\frac{d^{2}}{dx^{2}}$: $$f(x) = \int_{0}^{\infty}\{a(s)\cos(sx)+b(s)\sin(sx)\}ds$$ The problem was to find the coefficient functions $a(s)$ and $b(s)$ in terms of $f$. Then $-\frac{d^{2}}{dx^{2}}$ is formally turned into multiplication of the coefficient functions by $s^{2}$, i.e., $$-f''(x) = \int_{0}^{\infty}\{ s^{2}a(s)\cos(sx)+s^{2}b(s)\sin(sx)\}ds.$$ That's the idea behind most of the integral transforms: you start with a symmetric ordinary differential operator $Lf=-\frac{d}{dx}p\frac{d}{dx}f + qf$, you look for the eigenfunctions $Lf_{\lambda}=\lambda f_{\lambda}$ and you write a general $f$ as integral and/or discrete sums of the eigenfunctions $f_{\lambda}$, summing over $\lambda$. On old reference (out of print) written at the level of Advanced Calculus and dealing with general theory of integral transforms is R.V. Churchill's book listed below with an Amazon link.

R.V. Churchill, "Operational Mathematics": Amazon link

Overview of Heaviside's work, along with links to his publications: Heaviside Operator Calculus.
I highly recommend this person's web page; it's entertaining, informative, and has excellent references.

• Thanks for your reply. I am very interested in the book Operational Mathematics. But in my country it is not easy to get a copy of this book. Do you know where I can get a PDF version? Feb 22, 2016 at 3:28
• With some effort, I bought an used hard copy of Operational Mathematics from Amazon. I will read it. Apr 3, 2016 at 8:49
• @smwikipedia : Good. I don't know of any other works that deal with integral transforms in a more general way. And yet it is written to be used by undergraduates. Apr 4, 2016 at 3:19

Integral Transform is a huge subject. In my opinion, you should also have a strong background in Ordinary Differential Equation, Partial Differential Equation, and Real/Complex Analysis. Linear Algebra and Calculus is a "must" known subjects if you want to know where does Integral Transform come from. On the other hand, I found that Google and even Wikipedia don't have much information about integral transform in general, they just talk about some specifics topic such as Laplace, Fourier Transform, etc.There is a new book by K. Wolf from Springer and you should check it out, it's basic but deep enough in theory though. There is another book about ODE, but it has a section about Laplace Transform, and it's really details though not just a Transform table and showing you how to do it. I think it's a book by William A. Adskin.

• Could you tell me the name of the book by K.Worf from Springer? Thanks. Mar 13, 2015 at 8:58
• Integral Transforms in Science and Engineer Mar 13, 2015 at 17:28

As far as I know, there is no unified subject that specifically deals with integral transforms in general. Different integral transforms come up in different contexts. You are probably better off asking yourself, "What sort of engineering/mathematics do I want to study?" Depending on your answer, this may lead you to the study of some particular integral transforms.

Similarly, there are probably no books that deals with integral transforms generally - there are just too many to discuss. But you will find plenty of excellent books that discuss just a few transforms at a time, possibly, even focusing on just one. Some (maybe all) of the transforms you have listed have books entirely devoted to their study, and the Fourier transform essentially has an entire subfield of mathematics devoted to it. These should just be a Google search away.

As for the requisite knowledge, it depends on the particular transform you study, and in what depth. A solid background in calculus and linear algebra is definitely a must. If you go deep into the theory of such transforms, you will probably begin to encounter more sophisticated tools from real, complex, and functional analysis.