Improper Integrals

Complex Analysis · Axiom Academy

Integrals over the entire real line — solved by closing a contour up in the complex plane. An integral over the whole real line — and a trick to crack it Some of the most useful integrals in physics and engineering run over all of the real line, from - to + . Many of them have no elementary antiderivative — there is nothing to plug into the Fundamental Theorem. The residue theorem offers a stunning way out: lift the problem into the complex plane, close the path into a loop, and read the answer straight off the poles trapped inside. Watch the move. We start with the real-line piece from -R to R , then bend a giant semicircle over the top to seal it into a closed loop. By the residue theorem, the loop integral equals 2 i times the residues inside — here, the single pole at z=i . The closed loop hands you 2 i\, (f,i)=2 i (- 2 )= — and that is exactly _ - ^ x^2+1 . Why is the arc allowed to disappear? The whole trick only works if the giant arc contributes nothing as it grows. Drag R outward: the arc gets longer (its length is R ), but a function that decays like |f(z)| M/R^2 shrinks faster. Their product — the bound on the arc integral — collapses toward zero, and the real-line piece swallows more and more of (- , ) . When Q P + 2 , the bound M/R 0 : the arc vanishes, leaving _ - ^ f\,dx = f\,dz .

This is the written version of the interactive lesson above. See the full Complex Analysis course.