Residue theorem
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In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
s of
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem.


Statement

The statement is as follows: Let be a
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
open subset In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
of the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
containing a finite list of points , , and a function defined and holomorphic on . Let be a closed
rectifiable curve Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Rec ...
in , and denote the
winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of t ...
of around by . The line integral of around is equal to times the sum of residues of at the points, each counted as many times as winds around the point: \oint_\gamma f(z)\, dz = 2\pi i \sum_^n \operatorname(\gamma, a_k) \operatorname( f, a_k ). If is a positively oriented
simple closed curve In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an " interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exteri ...
, if is in the interior of , and 0 if not, therefore \oint_\gamma f(z)\, dz = 2\pi i \sum \operatorname( f, a_k ) with the sum over those inside . The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve 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 a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
on . Thus if two planar regions and of enclose the same subset of , the regions and lie entirely in , and hence \int_ d(f \, dz) - \int_ d(f \, dz) 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.


Examples


An integral along the real axis

The integral \int_^\infty \frac\,dx arises in
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
when calculating the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function ::\mathbf_A\colon X \to \, :which for a given subset ''A'' of ''X'', has value 1 at points ...
of the
Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...
. It resists the techniques of elementary
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
but can be evaluated by expressing it as a limit of contour integrals. Suppose and define the contour that goes along the real line from to and then counterclockwise along a semicircle centered at 0 from to . Take to be greater than 1, so that the imaginary unit is enclosed within the curve. Now consider the contour integral \int_C \,dz = \int_C \frac\,dz. Since is an
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
(having no singularities at any point in the complex plane), this function has singularities only where the denominator is zero. Since , that happens only where or . Only one of those points is in the region bounded by this contour. Because is \begin \frac & =\frac\left(\frac-\frac\right) \\ & =\frac -\frac , \end the residue of at is \operatorname_f(z)=\frac. According to the residue theorem, then, we have \int_C f(z)\,dz=2\pi i\cdot\operatorname\limits_f(z)=2\pi i \frac = \pi e^. The contour may be split into a straight part and a curved arc, so that \int_ f(z)\,dz+\int_ f(z)\,dz=\pi e^ and thus \int_^a f(z)\,dz =\pi e^-\int_ f(z)\,dz. Using some estimations, we have \left, \int_\frac\,dz\ \leq \pi a \cdot \sup_ \left, \frac \ \leq \pi a \cdot \sup_ \frac \leq \frac, and \lim_ \frac = 0. The estimate on the numerator follows since , and for complex numbers along the arc (which lies in the upper half-plane), the argument of lies between 0 and . So, \left, e^\ = \left, e^\=\left, e^\=e^ \le 1. Therefore, \int_^\infty \frac\,dz=\pi e^. If then a similar argument with an arc that winds around rather than shows that \int_^\infty\frac\,dz=\pi e^t, and finally we have \int_^\infty\frac\,dz=\pi e^. (If then the integral yields immediately to elementary calculus methods and its value is .)


An infinite sum

The fact that has simple poles with residue 1 at each integer can be used to compute the sum \sum_^\infty f(n). Consider, for example, . Let be the rectangle that is the boundary of with positive orientation, with an integer . By the residue formula, \frac \int_ f(z) \pi \cot(\pi z) \, dz = \operatorname\limits_ + \sum_^N n^. The left-hand side goes to zero as since the integrand has order O(n^). On the other hand,. Note that the Bernoulli number B_ is denoted by B_ in Whittaker & Watson's book. \frac \cot\left(\frac\right) = 1 - B_2 \frac + \cdots where the Bernoulli number B_2 = \frac. (In fact, .) Thus, the residue is . We conclude: \sum_^\infty \frac = \frac which is a proof of the Basel problem. The same trick can be used to establish the sum of the Eisenstein series: \pi \cot(\pi z) = \lim_ \sum_^N (z - n)^. We take with a non-integer and we shall show the above for . The difficulty in this case is to show the vanishing of the contour integral at infinity. We have: \int_ \frac \, dz = 0 since the integrand is an even function and so the contributions from the contour in the left-half plane and the contour in the right cancel each other out. Thus, \int_ f(z) \pi \cot(\pi z) \, dz = \int_ \left(\frac + \frac\right) \pi \cot(\pi z) \, dz goes to zero as .


See also

* Cauchy's integral formula * Glasser's master theorem *
Jordan's lemma In complex analysis, Jordan's lemma is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. The lemma is named after the French mathematician Camille Jordan. Statement Conside ...
* Methods of contour integration * Morera's theorem * Nachbin's theorem *
Residue at infinity In complex analysis, a branch of mathematics, the residue at infinity is a residue of a holomorphic function on an annulus having an infinite external radius. The ''infinity'' \infty is a point added to the local space \mathbb C in order to rend ...
*
Logarithmic form In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne. Let ''X'' be a complex manifold, ''D'' ⊂ ''X' ...


Notes


References

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External links

* {{springer, title=Cauchy integral theorem, id=p/c020900
Residue theorem
in MathWorld Theorems in complex analysis Analytic functions