the simple-pole residue formula

If you had to extract one Laurent coefficient by hand every time, residue calculus would be exhausting. Luckily, for the most common kind of bad point — a simple pole, where the function blows up like 1 / (z - z_0) and no worse — there is a one-line recipe that skips the series entirely.

The rule: if f has a simple pole at z_0, then Res(f, z_0) = limit as z approaches z_0 of (z - z_0) f(z). The idea is transparent. Near a simple pole the Laurent series is f(z) = a_{-1} / (z - z_0) + a_0 + a_1 (z - z_0) + .... Multiply through by (z - z_0): you get a_{-1} + a_0 (z - z_0) + a_1 (z - z_0)^2 + .... Now every term except a_{-1} contains a factor (z - z_0), so as z approaches z_0 they all vanish and only a_{-1} = Res(f, z_0) remains. Multiplying by (z - z_0) is exactly what cancels the singularity so a limit becomes possible.

This is the workhorse formula of the whole residue trade because simple poles are everywhere — rational functions with distinct roots in the denominator, trigonometric integrands, and Fourier integrals all serve them up. Just be sure the pole really is simple: if you apply this limit to a double pole, multiplying by only one factor of (z - z_0) leaves a 1 / (z - z_0) still blowing up, and the limit is infinite, which is the formula's honest way of telling you that you used the wrong recipe.