I am studying online course notes from University of Waterloo on 'Analytical mathematics in geology' in which the author describes a 'modified fourier transform' which can be used to incorporate 3rd kind of boundary conditions. The formula is(adsbygoogle = window.adsbygoogle || []).push({});

## \Gamma \small[ f(x) \small] = \bar{f}(a) = \int_{0}^{\infty } f(x) [a \cos(ax) + h\sin(ax)] dx ##

where $$ \Gamma $$ be the fourier operator.

with operational property

##\Gamma [\frac{d^2f}{dx^2}] = -a^2 \bar{f}(a) - a [ \frac{df}{dx}|_{x=0} - hf|_{x=0} ]##

I am trying to look for the detailed derivation of this equation in literature but have not found so far! Can anybody tell me in which book I can find detailed derivation of this formula and possibly with application?

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# A Fourier Transform for 3rd kind of boundary conditions?

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