- #1
sufive
- 19
- 0
As the title, I want to know details of the following integrations
\int |x|^a * exp[i*k*x] * dx = k^{-1-a} * Gamma[1+a] * sin[a*pi/2] -------(1)
by variable changes, k*x -> z, it's easy to get the factor k^{-1-a}, i.e.
l.h.s -> (\int z^a * exp[i*z] * dz) / k^{1+a} -------------------------------------(2)
but the remaining integration seems very difficult.
We know,
\int z^a * exp[-z] * dz \propto Gamma[1+a] --------------------------------------(3)
But, how to do integrations in eq(2) whose exponential argument is
imaginary instead negative?
\int |x|^a * exp[i*k*x] * dx = k^{-1-a} * Gamma[1+a] * sin[a*pi/2] -------(1)
by variable changes, k*x -> z, it's easy to get the factor k^{-1-a}, i.e.
l.h.s -> (\int z^a * exp[i*z] * dz) / k^{1+a} -------------------------------------(2)
but the remaining integration seems very difficult.
We know,
\int z^a * exp[-z] * dz \propto Gamma[1+a] --------------------------------------(3)
But, how to do integrations in eq(2) whose exponential argument is
imaginary instead negative?