Editing Residue (complex analysis) (section)

== Calculation of residues ==
Suppose a [[punctured disk]] ''D'' = {''z'' : 0 < |''z'' &minus; ''c''| < ''R''} in the complex plane is given and ''f'' is a [[holomorphic function]] defined (at least) on ''D''. The residue Res(''f'', ''c'') of ''f'' at ''c'' is the coefficient ''a''<sub>&minus;1</sub> of {{nowrap|(''z'' &minus; ''c'')<sup>&minus;1</sup>}} in the [[Laurent series]] expansion of ''f'' around ''c''. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.

According to the [[residue theorem]], we have:

: <math>\operatorname{Res}(f,c) = {1 \over 2\pi i} \oint_\gamma f(z)\,dz</math>

where ''γ'' traces out a circle around ''c'' in a counterclockwise manner and does not pass through or contain other singularities within it. We may choose the path ''γ'' to be a circle of radius ''ε'' around ''c.'' Since ''ε'' can be as small as we desire it can be made to contain only the singularity of c due to nature of isolated singularities. This may be used for calculation in cases where the integral can be calculated directly, but it is usually the case that residues are used to simplify calculation of integrals, and not the other way around.

===Removable singularities===
If the function ''f'' can be [[Analytic continuation|continued]] to a [[holomorphic function]] on the whole disk <math>|y-c|<R</math>, then Res(''f'',&nbsp;''c'')&nbsp;=&nbsp;0. The converse is not generally true.

===Simple poles===
If ''c'' is a [[simple pole]] of ''f'', the residue of ''f'' is given by:

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

If that limit does not exist, then ''f'' instead has an essential singularity at ''c''. If the limit is 0, then ''f'' is either analytic at ''c'' or has a removable singularity there. If the limit is equal to infinity, then the order of the pole is higher than 1.

It may be that the function ''f'' can be expressed as a quotient of two functions, <math>f(z)=\frac{g(z)}{h(z)}</math>, where ''g'' and ''h'' are [[holomorphic function]]s in a [[Neighbourhood (mathematics)|neighbourhood]] of ''c'', with ''h(c)'' = 0 and&nbsp;''h'(c)''&nbsp;≠&nbsp;0. In such a case, [[L'Hôpital's rule]] can be used to simplify the above formula to:

: <math>
\begin{align}
\operatorname{Res}(f,c) & =\lim_{z\to c}(z-c)f(z) = \lim_{z\to c}\frac{z g(z) - cg(z)}{h(z)} \\[4pt]
& = \lim_{z\to c}\frac{g(z) + z g'(z) - cg'(z)}{h'(z)} = \frac{g(c)}{h'(c)}.
\end{align}
</math>

===Limit formula for higher-order poles===
More generally, if ''c'' is a [[pole (complex analysis)|pole]] of order ''p'', then the residue of ''f'' around ''z'' = ''c'' can be found by the formula:

: <math> \operatorname{Res}(f,c) = \frac{1}{(p-1)!} \lim_{z \to c} \frac{d^{p-1}}{dz^{p-1}} \left( (z-c)^p f(z) \right). </math>

This formula can be very useful in determining the residues for low-order poles. For higher-order poles, the calculations can become unmanageable, and [[series expansion]] is usually easier. For [[essential singularity|essential singularities]], no such simple formula exists, and residues must usually be taken directly from series expansions.

===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>

=== Series methods ===
If parts or all of a function can be expanded into a [[Taylor series]] or [[Laurent series]], which may be possible if the parts or the whole of the function has a standard series expansion, then calculating the residue is significantly simpler than by other methods. The residue of the function is simply given by the coefficient of <math>(z-c)^{-1}</math> in the [[Laurent series]] expansion of the function.