Editing Residue (complex analysis) (section)

===Residue at infinity===
In general, the [[residue at infinity]] is defined as:

: <math> \operatorname{Res}(f(z), \infty) = -\operatorname{Res}\left(\frac{1}{z^2} f\left(\frac 1 z \right), 0\right).</math>

If the following condition is met:

:<math> \lim_{|z| \to \infty} f(z) = 0,</math>

then the [[residue at infinity]] can be computed using the following formula:

:<math> \operatorname{Res}(f, \infty) = -\lim_{|z| \to \infty} z \cdot f(z).</math>

If instead

:<math> \lim_{|z| \to \infty} f(z) = c \neq 0,</math>

then the [[residue at infinity]] is

:<math> \operatorname{Res}(f, \infty) = \lim_{|z| \to \infty} z^2 \cdot f'(z).</math>
:

For functions meromorphic on the entire complex plane with finitely many singularities, the sum of the residues at the (necessarily) isolated singularities plus the residue at infinity is zero, which gives:

: <math> \operatorname{Res}(f(z), \infty) = -\sum_k \operatorname{Res} (f(z), a_k).</math>