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Mathematical Methods of Physics: For Undergraduate

Sunil Karna

Chapter 6 Integral Transform

An integral transform is a mathematical operation that transforms a function into another function through the use of an integral. It is a tool used to solve differential equations and some analyze functions. There are several important integral transforms, each with its own properties and applications. Some of the commonly used integral transforms include: Laplace Transform: The Laplace transform converts a function of time into a function of a complex variable. It is particularly useful for solving ordinary and partial differential equations, analyzing linear time-invariant systems, and studying transient behavior. Fourier Transform: The Fourier transform converts a function of time into a function of frequency. It decomposes a function into its constituent frequency components and is extensively used in signal processing, image processing, and various other fields. There are various transforms used in practice for various purposes.
The solution of differential equations with boundary value can easily be determined with the help of integral transforms. The integral transform \(F(s)\) of a function \(f(x)\) is defined as
\begin{equation*} \mathscr{F}\{f(x)\}=F(s) = \int\limits_{a}^{b}f(x)k(s,x)\,dx \end{equation*}
where \(k(s,x)\) is called a kernal of the function. For example,
  1. Fourier transform
    1. Fourier complex transform \(k(s,x) = e^{-isx}\text{.}\)
      \begin{equation*} \mathscr{F}\{f(x)\}=F(s) = \int\limits_{-\infty}^{\infty}f(x) e^{-isx}\,dx \end{equation*}
    2. Fourier sine transform \(k(s,x) = \sin sx\text{.}\)
      \begin{equation*} \mathscr{F}_{s}\{f(x)\}=F(s) = \int\limits_{0}^{\infty}f(x) \sin sx\,dx \end{equation*}
    3. Fourier cosine transform \(k(s,x) = \cos sx\text{.}\)
      \begin{equation*} \mathscr{F}_{c}\{f(x)\}=F(s) = \int\limits_{0}^{\infty}f(x) \cos sx\,dx \end{equation*}
  2. Laplace transform \(k(s,x) = e^{-sx}\text{.}\)
    \begin{equation*} \mathscr{L}\{f(x)\}=F(s) = \int\limits_{0}^{\infty}f(x) e^{-sx}\,dx \end{equation*}
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