The discussion revolves around evaluating an improper integral using the Fourier Integral Theorem. The specific integral in question involves the cosine function and a parameter alpha, with the goal of demonstrating a relationship to an exponential function.
Some participants have shared their initial attempts at expressing the cosine function, while others are exploring the definition and implications of the Fourier Integral Theorem. There is an ongoing exploration of methods, but no consensus or resolution has been reached yet.
Participants note the requirement to evaluate the integral directly using the Fourier Integral Theorem, which raises questions about the theorem's specifics and how it applies to the given problem.
Fusiontron said:Homework Statement
Show that
integral from 0 - > infinity (cos(alpha*x)/(alpha^2 + 1))dalpha = (pi/2)exp(-x)
Homework Equations
The Attempt at a Solution
cos(alpha*x) = (1/2)(exp(i*alpha*x)+exp(-i*alpha*x))
Really don't know where to go from here.
Fusiontron said:The problem says to evaluate directly with the Fourier Integral Theorem.