Let be a
simply connectedopen subset of the
complex plane containing a finite list of points and a function holomorphic on Letting be a closed
rectifiable curve in and denoting the
residue of at each point by and the
winding number of around by the line integral of around is equal to times the sum of residues, each counted as many times as winds around the respective point:
The relationship of the residue theorem to Stokes' theorem is given by the
Jordan curve theorem. The general
plane curveγ must first be reduced to a set of simple closed curves whose total is equivalent to for integration purposes; this reduces the problem to finding the integral of along a Jordan curve with interior The requirement that be holomorphic on is equivalent to the statement that the
exterior derivative on Thus if two planar regions and of enclose the same subset of the regions and lie entirely in hence
is well-defined and equal to zero. Consequently, the contour integral of along is equal to the sum of a set of integrals along paths each enclosing an arbitrarily small region around a single — the residues of (up to the conventional factor at Summing over we recover the final expression of the contour integral in terms of the winding numbers
In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed (which is usually easy), and a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle. The integral over this curve can then be computed using the residue theorem. Often, the half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were originally interested in.
Suppose a
punctured diskD = {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. 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
continued to a
holomorphic function on the whole disk , then Res(f, c) = 0. The converse is not generally true.
If that limit does not exist, there is an essential singularity there. If it is 0 then it is either analytic there or there is a removable singularity. If it is equal to infinity then the order is higher than 1.
It may be that the function f can be expressed as a quotient of two functions, , where g and h are
holomorphic functions in a
neighbourhood of c, with h(c) = 0 and h(c) ≠ 0. In such a case,
L'Hôpital's rule can be used to simplify the above formula to:
Limit formula for higher-order poles
More generally, if c is a
pole of order n, 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.
For holomorphic functions the sum of the residues at the isolated singularities plus the residue at infinity is zero which gives:
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 in the
Laurent series expansion of the function.
Suppose t > 0 and define the contour C that goes along the
real line from −a to a and then counterclockwise along a semicircle centered at 0 from a to −a. Take a to be greater than 1, so that the
imaginary unit i is enclosed within the curve. Now consider the contour integral
Since eitz is an
entire function (having no
singularities at any point in the complex plane), this function has singularities only where the denominator z2 + 1 is zero. Since z2 + 1 = (z + i)(z − i), that happens only where z = i or z = −i. Only one of those points is in the region bounded by this contour. Because f(z) is
The estimate on the numerator follows since t > 0, and for
complex numbersz along the arc (which lies in the upper half-plane), the argument φ of z lies between 0 and π. So,
Therefore,
If t < 0 then a similar argument with an arc C′ that winds around −i rather than i shows that
and finally we have
(If t = 0 then the integral yields immediately to elementary calculus methods and its value is π.)
Evaluating zeta functions
The fact that π cot(πz) has simple poles with residue 1 at each integer can be used to compute the sum
Consider, for example, f(z) = z−2. Let ΓN be the rectangle that is the boundary of [−N − 1/2, N + 1/22 with positive orientation, with an integer N. By the residue formula,
The left-hand side goes to zero as N → ∞ since is uniformly bounded on the contour, thanks to using on the left and right side of the contour, and so the integrand has order over the entire contour. On the other hand,[2]
Lindelöf, Ernst L. (1905). Le calcul des résidus et ses applications à la théorie des fonctions (in French). Editions Jacques Gabay (published 1989).
ISBN2-87647-060-8.
Mitrinović, Dragoslav; Kečkić, Jovan (1984). The Cauchy method of residues: Theory and applications. D. Reidel Publishing Company.
ISBN90-277-1623-4.