The integral transforms considered thus far are applicable to problems involving either semiinfinite or infinite domains. However, in applying the method of integral transforms to problems formulated on finite domains it is necessary to introduce finite intervals on the transform integral. We then find that it is possible to derive their inverses from the theory of Fourier series. Transforms of this nature are called finite transforms and sometimes afford a more convenient method of solution than the classical methods which often require much ingenuity in assuming at the outset a correct solution form.
In this chapter we will introduce finite Fourier transforms and the finite Hankel transform, the latter being a special case of the more general Sturm-Liouville transform.
Let us begin by considering the simplest cases of finite transforms, which are known as finite sine or cosine transforms. The general theory of these transforms is based on the theory of Fourier series, with which we assume the reader is familiar.
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