The residue theorem in complex analysis is a powerful tool to evaluate path integrals of meromorphic functions over closed curves and can often be used to compute real integrals as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. Jump to: navigation, search This article may be too technical for most readers to understand. ... In mathematics, a path integral (also known as a line integral) is an integral where the function to be integrated is evaluated along a path or curve. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles fore the function. ... In mathematics, the Cauchy integral theorem in complex analysis, named after Augustin Louis Cauchy, is an important statement about path integrals for holomorphic functions in the complex plane. ... In mathematics, Cauchys integral formula, named after Augustin Louis Cauchy, is a central statement in complex analysis. ...

The statement is as follows. Suppose U is a simply connected open subset of the complex plane C, a₁,...,a_n are finitely many points of U and f is a function which is defined and holomorphic on U {a₁,...,a_n}. If γ is a rectifiable curve in U which doesn't meet any of the points a_k and whose start point equals its endpoint, then A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... Jump to: navigation, search In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ...

Here, Res(f, a_k) denotes the residue of f at a_k, and I(γ, a_k) is the winding number of the curve γ about the point a_k. This winding number is an integer which intuitively measures how often the curve γ winds around the point a_k; it is positive if γ moves in a counter clockwise ("mathematically positive") manner around a_k and 0 if γ doesn't move around a_k at all. In complex analysis, the residue is a complex number which describes the behavior of path integrals of a meromorphic function around a singularity. ... A point z0 and a curve C In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. ... Jump to: navigation, search The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

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. 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 if it is large enough, leaving only the real-axis part of the integral, the one we were originally interested in.

Example

The integral

(which arises in probability theory as (a scalar multiple of) the characteristic function of the Cauchy distribution) resists the techniques of elementary calculus. We will evaluate it by expressing it as a limit of contour integrals along 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. The contour integral is Image File history File links Contour Diagram File links The following pages link to this file: Residue theorem Methods of contour integration User:MaxPower/myImages ... Probability theory is the mathematical study of probability. ... Some mathematicians use the phrase characteristic function synonymously with indicator function. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... Jump to: navigation, search Calculus is a central branch of mathematics, developed from algebra and geometry, and built on two major complementary ideas. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, an imaginary number (or purely imaginary number) is a complex number whose square is a negative real number or zero. ...

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 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 In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ... In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability. ...

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the residue of f(z) at z = i is In complex analysis, the residue is a complex number which describes the behavior of path integrals of a meromorphic function around a singularity. ...

According to the residue theorem, then, we have

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

∫	+	∫	= πe − t,
straight		arc

and thus

It can be shown that if t > 0 then

Therefore if t > 0 then

A similar argument with an arc that winds around −i rather than i shows that if t < 0 then

and finally we have

(If t = 0 then the integral yields immediately to first-year calculus methods and its value is π.)