What is residue?

Example 17.7 (The four classical integral types)

(a) Rational over $\R$: for $\int_\R\frac{\dd x}{1 + x^4}$, close with a large semicircle $S_R$ in the upper half-plane: the integrand is $O(R^{-4})$ there, so $\int_{S_R} \to 0$ (ML), and the residue theorem with the poles $\eu^{\iu\pi/4}, \eu^{3\iu\pi/4}$ (simple, residues $\frac1{4z^3} = \frac{z}{4z^4} = -\frac z4$ at a pole) gives

(b) Fourier type: for $t \geq 0$, $\int_\R\frac{\eu^{\iu tx}}{1 + x^2}\dd x = 2\iu\pi\operatorname{Res}\bigl(\tfrac{\eu^{\iu tz}}{1+z^2}, \iu\bigr) = 2\iu\pi\frac{\eu^{-t}}{2\iu} = \pi\eu^{-t}$ — the upper semicircle works because $\abs{\eu^{\iu tz}} = \eu^{-t\operatorname{Im}z} \leq 1$ there; taking real parts: $\int_\R\frac{\cos(tx)}{1+x^2}\dd x = \pi\eu^{-\abs t}$, settling Exercise 10.10’s admitted formula. (c) Trigonometric over a period: substitute $z = \eu^{\iu t}$, $\cos t = \frac{z + z^{-1}}2$, $\dd t = \frac{\dd z}{\iu z}$: $\int_0^{2\pi}\frac{\dd t}{a + \cos t}$ ($a > 1$) becomes a residue count inside the unit circle (Exercise 17.2). (d) Series: pair $f$ with $\pi\cot(\pi z)$, whose poles are the integers with residue $1$: the weekend problem sums $\sum n^{-2}$ and $\sum n^{-4}$ this way.