Derivation of the Fourier-Mellin Inversion Theorem

We recall from Section 10.5.1 in Chapter 10, that solutions of the Sturm-LiouvUle equation, along with suitable Sturm-Uouville boundary conditions, always produced orthogonal functions. Thus, the functions 

if we represent a function f(,x) in this interval by an expansion of such orthogonal functions 

The series representing fix) is called a Fourier-Sine series. Similarly, we can also express functions in terms of the Fourier-Cosine series 

along with C13, are known as the Fourier-Cosine representation of fix), in the interval 0 jc L (Hildebrand 1%2). 

The question arises, how is it possible to obtain a similar representation in the complete semi-infinite interval with x 0 It is clear if we replaced L with in 

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APPENDIX C Derivation of the Fourier-Mellin Inversion Theorem written for the problem at hand as... 

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