In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.
that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.)
Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.
has an analytic antiderivative in a punctured disk
Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a−1 of a Laurent series.
The concept can be used to provide contour integration values of certain contour integral problems considered in the residue theorem.
is a positively oriented simple closed curve around
and not including any other singularities on or inside the curve.
The definition of a residue can be generalized to arbitrary Riemann surfaces.
Since path integral computations are homotopy invariant, we will let
If a function is expressed as a Laurent series expansion around c as follows:
around a point c. Hence, if a Laurent series representation of a function exists around c, then its residue around c is known by the coefficient of the
, with a finite set of singularities within a positively oriented simple closed curve
which does not pass through any singularity, the value of the contour integral is given according to residue theorem, as:
Suppose a punctured disk D = {z : 0 < |z − 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−1 of (z − c)−1 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: 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.
If c is a simple pole of f, the residue of f is given by: 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.
In such a case, L'Hôpital's rule can be used to simplify the above formula to: More generally, if c is a pole of order p, then the residue of f around z = c can be found by the formula: 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 singularities, no such simple formula exists, and residues must usually be taken directly from series expansions.
In general, the residue at infinity is defined as: If the following condition is met: then the residue at infinity can be computed using the following formula: If instead then the residue at infinity is 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: 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.
As an example, consider the contour integral where C is some simple closed curve about 0.
The integral then becomes Let us bring the 1/z5 factor into the series.
The series of the path integrals then collapses to a much simpler form because of the previous computation.
Recall the expression for the Taylor series for a function g(z) about z = a:
The next example shows that, computing a residue by series expansion, a major role is played by the Lagrange inversion theorem.
because the first series converges uniformly on any small circle around 0.
Note that, with the corresponding stronger symmetric assumptions on