Integral Transforms: Origins & Derivations

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In summary, the conversation discusses the concept of integral transforms and their various properties and uses. The LaPlace transform and the Fourier transform are the most widely used and have specific properties that make them useful for certain types of problems. The derivation of these transforms is not accidental, but rather a result of manipulating Taylor series and operational properties. There are also other transforms that are related to these two and have limited applicability. Online resources for understanding the derivation and need for integral transforms can be found in books and other materials such as "Mathews & Walker - Mathematical methods of Physics".
  • #1
amcavoy
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I am curious about the different integral transforms. I found definitions online, but where did they come from? Does anyone know a good site that shows the derivation / need for the common integral transforms?

Thanks.
 
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  • #2
The LaPlace transform is an integral transform:
[tex]L(f)= \int_0^{\infty}e^{-st}f(t)dt[/tex]
which has the nice property of transforming (linear) differential equations to algebraic equations.

The Fourier transform:
[tex] F(f)= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-itx}f(t)dt[/tex]
is a generalization of Fourier series to infinite intervals.
 
  • #3
Alright. So take, for instance, the Laplace Transform. Was it stumbled upon by accident, or is there an actual derivation for it?
 
  • #4
Read "Mathews & Walker - Mathematical methods of Physics"
 
  • #5
Are there any online resources?

Thanks again.
 
  • #6
An Integral transform is a linear transform of the form.
[tex]L[f(t)]:=\int_a^bK(s,t)f(t)dt[/tex]
where a and b may be + or - infinity and K is called the kernel.
Integral transform kernels are selected using three main traits.
1. Existence/convergence
For functions under consideration does the transform exist? What about inversion? Do integrals that arise converge? In some cases the definition of integral is expanded using Cauchy principle values or distribution theory.
2. Computability
Can the transform be found analytically or numerically? What about inversion? For instance the Laplace, and Fourier transforms are often computed analytically. Numerical inversion of Laplace transform is difficult. The Hankel and Fourier transforms are often inverted numerically.
3. Operational properties
Do functions operated on by some operator have transforms that are easy to deal with?
In practice not many Transforms are of wide applicability they are
-Laplace transform and variations. Useful for temporal problems.
-Fourier transform and variations. Useful for spatial problems.
-Special transforms. Useful in very limited situations. Things like Greens functions that are often useful for only one specific problem.
apmcavoy said:
Alright. So take, for instance, the Laplace Transform. Was it stumbled upon by accident, or is there an actual derivation for it?
Derivation is I think the wrong choice of words. One can be led to the transform. The first way is as a variation on Taylors series.
[tex]\sum_{n=0}^{\infty} a_ne^{-n s}\Delta n[/tex]
This is Taylors series in a new form that leads the way to the Laplace transform. Delta(n)=1 so it is normaly omited, and normally z is used instead of exp(-s).
Now we allow n to be a sequence rather then just integers
[tex]\sum_{n=0}^{\infty} a_ne^{-\lambda_n s}\Delta\lambda_n[/tex]
this is a Riemann sum so it is natural to take the limit to an integral
[tex]\int_0^{\infty}f(t)e^{-st}dt[/tex]
A similar process leads from the Fourier trig series to the Fourier transform.
[tex]\sum_{n=-\infty}^{\infty} a_ne^{i n s}\Delta n[/tex]
allow nonintegers
[tex]\sum_{n=-\infty}^{\infty} a_ne^{i\lambda_n s}\Delta\lambda_n[/tex]
Limit to integral
[tex]\int_{-\infty}^{\infty}f(t)e^{ist}dt[/tex]
We may also be led to the Laplace transform by operational properties. We desire a kernel so that
[tex]\int_0^{\infty}K(s,t)f'(t)dt=s\int_0^{\infty}K(s,t)f(t)dt[/tex]
integration by parts on the left side leads us to
[tex]K(s,t)f(t)|_{t=0}^{t=\infty}=\int_0^{\infty}(K'(s,t)+sK(s,t))f(t)dt[/tex]
the ' notation means t partial
Now we make things simple by requiring the left side to be zero. Thus the right side is zero giving a differential equation
K'(s,t)+sK(s,t)=0
the solution is
K(s,t)=A(s)exp(-s*t)
where A(s) is an abitrary function of s
The standard choice is A(s)=1 to make things nice.
The second most popular choice used in some older work is the p-multiplied form where A(s)=s.
Thus we are again led to the Laplace transform.
Now I will list several transforms they are related to one another, so we really have few. Also there are many more than these, but they are mostly either variations of these, or of quite limited usefulness.
-Laplace transform
-p-multiplied Laplace transform (Laplace multiplies by the variable)
-Bilateral Laplace Transform (Laplace with lower limit -infinity)
-Stiltjes transform (Laplace twice)
-Fourier transform
-Fourier cos transform (real part of fourier)
-Fourier sin transform (imaginary part of fourier)
-Hartley transform (F. cos transform - f. sin transform a Fourier like real transform)
-various Finite range Fourier transforms (any Fourier like transform on a finite interval)
-Hilbert transform (similar to Stiltjes the principle value of a singular integral)
-Hankel transform (involves bessel functions like a Fourier transform for polar coordinates)
-various transforms that are like Fourier for some coordinate system.
 

1. What is an integral transform?

An integral transform is a mathematical operation that converts a function from one domain to another by using an integral. It is used to solve differential equations and to express functions in a more convenient form for analysis.

2. What are some common examples of integral transforms?

Some common examples of integral transforms include the Fourier transform, Laplace transform, and Z-transform. These transforms are widely used in physics, engineering, and mathematics to solve problems involving differential equations and signals.

3. What is the origin of integral transforms?

The concept of integral transforms can be traced back to the 19th century, with the work of mathematicians such as Joseph Fourier and Pierre-Simon Laplace. However, the modern theory of integral transforms was developed in the 20th century by mathematicians and physicists, with contributions from names such as Henri Poincaré, Norbert Wiener, and Jean Leray.

4. How are integral transforms derived?

The derivation of integral transforms involves using mathematical techniques such as integration and complex analysis. The specific method of derivation depends on the type of transform being used. Generally, the goal is to find a new representation of a function that simplifies its analysis or leads to a solution of a problem.

5. What are the applications of integral transforms?

Integral transforms have a wide range of applications in physics, engineering, and mathematics. They are used to solve differential equations, analyze signals, and study systems in various fields such as electromagnetics, quantum mechanics, and control theory. They also have applications in image and signal processing, data compression, and digital communication.

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