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In complex analysis, a discipline within mathematics, the **residue theorem**, sometimes called **Cauchy's residue theorem**, is a powerful tool to evaluate line integrals of analytic functions 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.

The statement is as follows:

Let U be a simply connected open subset of the complex plane containing a finite list of points *a*_{1}, ..., *a*_{n}, *U*_{0} = *U* \ {*a*_{1}, ..., *a*_{n}}, and a function f defined and holomorphic on *U*_{0}. Let γ be a closed rectifiable curve in *U*_{0}, and denote the winding number of γ around *a*_{k} by I(*γ*, *a*_{k}). The line integral of f around γ is equal to 2π*i* times the sum of residues of f at the points, each counted as many times as γ winds around the point:

If γ is a positively oriented simple closed curve, I(*γ*, *a*_{k}) = 1 if *a*_{k} is in the interior of γ, and 0 if not, therefore

with the sum over those *a*_{k} inside γ.^{ [1] }

The relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem. The general plane curve γ must first be reduced to a set of simple closed curves {*γ*_{i}} whose total is equivalent to γ for integration purposes; this reduces the problem to finding the integral of *f**dz* along a Jordan curve *γ*_{i} with interior V. The requirement that f be holomorphic on *U*_{0} = *U* \ {*a*_{k}} is equivalent to the statement that the exterior derivative *d*(*f**dz*) = 0 on *U*_{0}. Thus if two planar regions V and W of U enclose the same subset {*a*_{j}} of {*a*_{k}}, the regions *V* \ *W* and *W* \ *V* lie entirely in *U*_{0}, and hence

is well-defined and equal to zero. Consequently, the contour integral of *f**dz* along *γ*_{j} = ∂V is equal to the sum of a set of integrals along paths *λ*_{j}, each enclosing an arbitrarily small region around a single *a*_{j} — the residues of f (up to the conventional factor 2π*i*) at {*a*_{j}}. Summing over {*γ*_{j}}, we recover the final expression of the contour integral in terms of the winding numbers {I(*γ*, *a*_{k})}.

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.

The integral

arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of contour integrals.

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 *e*^{itz} is an entire function (having no singularities at any point in the complex plane), this function has singularities only where the denominator *z*^{2} + 1 is zero. Since *z*^{2} + 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 residue of *f*(*z*) at *z* = *i* is

According to the residue theorem, then, we have

The contour C may be split into a straight part and a curved arc, so that

and thus

Using some estimations, we have

and

The estimate on the numerator follows since *t* > 0, and for complex numbers z along the arc (which lies in the upper halfplane), 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.)

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 Γ_{N} be the rectangle that is the boundary of [−*N* − 1/2, *N* + 1/2]^{2} with positive orientation, with an integer N. By the residue formula,

The left-hand side goes to zero as *N* → ∞ since the integrand has order . On the other hand,^{ [2] }

- where the Bernoulli number

(In fact, *z*/2 cot(*z*/2) = *iz*/1 − *e*^{−iz} − *iz*/2.) Thus, the residue Res_{z=0} is −*π*^{2}/3. We conclude:

which is a proof of the Basel problem.

The same trick can be used to establish the sum of the Eisenstein series:

We take *f*(*z*) = (*w* − *z*)^{−1} with w a non-integer and we shall show the above for w. The difficulty in this case is to show the vanishing of the contour integral at infinity. We have:

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,

goes to zero as *N* → ∞.

- ↑ Whittaker & Watson 1920 , p. 112, §6.1.
- ↑ Whittaker & Watson 1920 , p. 125, §7.2. Note that the Bernoulli number is denoted by in Whittaker & Watson's book.

The **Cauchy distribution**, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the **Lorentz distribution**, **Cauchy–Lorentz distribution**, **Lorentz(ian) function**, or **Breit–Wigner distribution**. The Cauchy distribution is the distribution of the x-intercept of a ray issuing from with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.

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In mathematics, the **Cauchy integral theorem** in complex analysis, named after Augustin-Louis Cauchy, is an important statement about line integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a function is holomorphic everywhere in between the two paths, then the two path integrals of the function will be the same.

In mathematics, **Cauchy's integral formula**, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.

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. Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

In mathematics, the **digamma function** is defined as the logarithmic derivative of the gamma function:

In complex analysis, a branch of mathematics, **Morera's theorem**, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.

In the mathematical field of complex analysis, **contour integration** is a method of evaluating certain integrals along paths in the complex plane.

In complex analysis, the **argument principle** relates the difference between the number of zeros and poles of a meromorphic function to a contour integral of the function's logarithmic derivative.

In mathematics, there are several integrals known as the **Dirichlet integral**, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:

In mathematics, **holomorphic functional calculus** is functional calculus with holomorphic functions. That is to say, given a holomorphic function *f* of a complex argument *z* and an operator *T*, the aim is to construct an operator, *f*(*T*), which naturally extends the function *f* from complex argument to operator argument. More precisely, the functional calculus defines a continuous algebra homomorphism from the holomorphic functions on a neighbourhood of the spectrum of *T* to the bounded operators.

In mathematics, the **Nørlund–Rice integral**, sometimes called **Rice's method**, relates the *n*th forward difference of a function to a line integral on the complex plane. It commonly appears in the theory of finite differences and has also been applied in computer science and graph theory to estimate binary tree lengths. It is named in honour of Niels Erik Nørlund and Stephen O. Rice. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying saddle-point techniques to its evaluation.

In mathematics, in the area of complex analysis, **Nachbin's theorem** is commonly used to establish a bound on the growth rates for an analytic function. This article provides a brief review of growth rates, including the idea of a **function of exponential type**. Classification of growth rates based on type help provide a finer tool than big O or Landau notation, since a number of theorems about the analytic structure of the bounded function and its integral transforms can be stated. In particular, Nachbin's theorem may be used to give the domain of convergence of the **generalized Borel transform**, given below.

In complex analysis, **Jordan's lemma** is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. It is named after the French mathematician Camille Jordan.

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, although that is typically reserved for line integrals in the complex plane.

In probability theory and directional statistics, a **wrapped probability distribution** is a continuous probability distribution that describes data points that lie on a unit *n*-sphere. In one dimension, a wrapped distribution will consist of points on the unit circle. If φ is a random variate in the interval (-∞,∞) with probability density function *p(φ)*, then *z = e ^{ i φ}* will be a circular variable distributed according to the wrapped distribution

In mathematics, **Ramanujan's master theorem** is a technique that provides an analytic expression for the Mellin transform of an analytic function.

In mathematics, the **Abel–Plana formula** is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It states that

The **Egorychev method** is a collection of techniques for finding identities among sums of binomial coefficients. The method relies on two observations. First, many identities can be proved by extracting coefficients of generating functions. Second, many generating functions are convergent power series, and coefficient extraction can be done using the Cauchy residue theorem. The sought-for identity can now be found using manipulations of integrals. Some of these manipulations are not clear from the generating function perspective. For instance, the integrand is usually a rational function, and the sum of the residues of a rational function is zero, yielding a new expression for the original sum. The residue at infinity is particularly important in these considerations.

- Ahlfors, Lars (1979).
*Complex Analysis*. McGraw Hill. ISBN 0-07-085008-9. - Lindelöf, Ernst L. (1905).
*Le calcul des résidus et ses applications à la théorie des fonctions*(in French). Editions Jacques Gabay (published 1989). ISBN 2-87647-060-8. - Mitrinović, Dragoslav; Kečkić, Jovan (1984).
*The Cauchy method of residues: Theory and applications*. D. Reidel Publishing Company. ISBN 90-277-1623-4. - Whittaker, E. T.; Watson, G. N. (1920).
*A Course of Modern Analysis*(3rd ed.). Cambridge University Press.

- "Cauchy integral theorem",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Residue theorem in MathWorld

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