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### What is a Functional Transformation?

```Date: 08/11/2002 at 00:17:37
From: Britney
Subject: Definition of a functional transformation

Hi Dr. Math,

I would like a brief definition of what a functional transformation is
and what an application is in early Calculus.

Thank you in advance.
```

```
Date: 08/11/2002 at 16:14:49
From: Doctor Jaime
Subject: Re: Definition of a functional transformation

Hello,

A "functional transformation" or a "functional operator" is just a map
(function) whose domain is some set of functions (e.g. all functions
defined in the interval [0,2], all continuous functions, all real
functions with the same domain D). The range can be a set of numbers,
a set of functions or some other set.

So the difference between a "functional transformation" and maps you
have already studied is that the domain is a bit more complicated:
functions acting on functions.

Confused? You have already studied a "functional transformation" but
never gave it that name: the derivative. In fact a derivative is
applied to functions and the result is another function, the
derivative function. That's why this operation is usually called the
"differential operator":

Differential Operator - MathWorld
http://mathworld.wolfram.com/DifferentialOperator.html

I don't know if you have already studied integrals, but the definite
integral, the indefinite integral and the improper integral are also
examples of functional transformations, usually called "integral
transforms":

Integral Transform - MathWorld
http://mathworld.wolfram.com/IntegralTransform.html

You can study properties of these functional transformations, like
linearity. If they are linear they are called "linear
transformations":

Linear Transformation - MathWorld
http://mathworld.wolfram.com/LinearTransformation.html

Some famous examples of functional transformation are the Laplace
transformation and the Fourier transformation, which are intensively
used in physics and engineering. You can read something about them
here:

Laplace Transform - MathWorld
http://mathworld.wolfram.com/LaplaceTransform.html

Fourier Transform - MathWorld
http://mathworld.wolfram.com/FourierTransform.html

Please feel free to ask again if you have more questions. And don't
forget to visit our FAQ at

http://mathforum.com/dr.math/faq/

- Doctor Jaime, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Calculus
College Definitions
High School Calculus
High School Definitions
High School Functions

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