  Stieltjes integral theorem is more commonly known by the phrase 'integration
by parts' and enables rearrangement of an otherwise intractable integral to a
more amenable form; often permitting completion of an integral in closed form.
Applying Stieltjes integral theorem to the Hilbert transform yields an
equivalent alternate integral definition, which is homeomorphic and exhibits
accelerated computation. By virtue of the convolution theorem, the integral is
mapped to Fourier image space and delineates requirements for the inverse
Fourier transform, also, these requirements reveal the underlying reason for
increased computational speed. Lastly, an alternative to Cauchy's integral
formula is deduced by extending the line integral into the complex domain and
involves a line integral with logarithmic kernel.
