Residue at Simple Pole Example

Complex Analysis · Axiom Academy

EXAMPLE Residue at a Simple Pole Computing residues at order-1 poles with both the limit and the quotient formula Two formulas for a simple pole Nice work. You found both residues of a rational function — and saw the limit formula and the quotient shortcut land on the same number. Two equivalent tools. At a simple pole z_0 , either take the limit , or — when f=g/h with h(z_0)=0 but — use the shortcut . The shortcut skips the algebra of cancelling the factor. Use h' , never h . The denominator is zero at the pole, so g(z_0)/h(z_0) is undefined — the quotient rule divides by the derivative h'(z_0) . The same recipe handles complex and transcendental poles. , , and . The "residues sum to zero" check has a condition. It needs . Here because the degree gap is only 1 ; for the gap is 3 and the residues do sum to 0 . These single numbers are exactly what the residue theorem adds up: once you can extract the residue at each pole, a contour integral collapses to times their sum.

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