Graduate Fourier Transform for 3rd kind of boundary conditions?

[Atr cheema]

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

 

[eys_physics]

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

The formula is straightforward to derive it by adopting your definition of $\Gamma$ for $d^2/dx^2$ and using integration by parts twice. How you tried to derive the formula yourself?