for symmetry, though in many applications, the exact value at the point of discontinuity doesn't affect the integral transform. The Challenge of Direct Integration
Consider the modified function: $$ u(t)e^-\sigma t $$ fourier transform of heaviside step function
We can write the Heaviside function as: $$ u(t) = \frac12 + \frac12\textsgn(t) $$ for symmetry, though in many applications, the exact