Evaluating Real Integrals by Residues
Close a real integral into a contour in the plane, discard the added arc with an estimate, and read the answer off the residues.
What a learner can do afterwards
- Evaluate a rational real integral over the whole line by closing in a half-plane
- Handle an oscillatory integrand with Jordan's lemma and justify discarding the arc
- Deal with a pole on the contour by indenting around it and accounting for the half residue
1 · Read
Real integrals over the whole line become loop integrals by adding a giant semicircle. For rational functions decaying fast enough, the arc share dies as the radius grows, leaving 2 pi i times the upper half plane residues. Decay is the whole game: the denominator must outgrow the numerator by at least two degrees. Check this before closing.
Sines and cosines need Jordan lemma. Write the cosine as the real part of e to the i a z, close upwards for positive frequency, and the upstairs exponential decay kills the arc. Take the real part at the end. The sign picks the half plane: positive frequencies close up, negative ones close down. The wrong side makes the arc explode.
Poles sitting on the path need indenting. Detour around the pole with a tiny semicircle, which in the limit contributes half a residue, positive or negative by the detour side. The straight runs become a principal value integral. Sketch the detour and its orientation with care, since one sign slip ruins the sum.
Close with intent, not hope. Name the half plane, verify the decay, list enclosed poles, then treat boundary poles by indenting. Never count a pole you did not enclose.
Close the line into a loop, kill the arc, and let enclosed residues answer.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.