Residue theorem

The residue theorem is an important theorem of function theory, a sub- area of mathematics. It represents a generalization of the Cauchy integral theorem and Cauchy's integral formula dar. His importance resides not only in the far-reaching consequences in the theory of functions, but also in the practical calculation of real integrals.

He states that the line integral along a closed curve depends on a holomorphic up isolated singularities function only from residual in the singularities in the interior of the curve and the number of turns in the curve of these singularities. Instead of a line integral so you need only calculate residuals and number of turns, which is easier in many cases.

Set

Be an elementary domain, ie a simply connected domain in the complex plane. Furthermore, let a discrete subset of a holomorphic function, a real interval and a closed path in. Then for the complex path integral

Where the winding number of with respect to and the residue of in is.

Comments

• The sum on the right side is always finite, because of the enclosed ( simply connected ) region is relatively compact in and thus limited. Because is discrete in, is finite, and only these are the points that contribute to the sum, as for all other the number of turns or the residue disappears.
• If it is at the points in order liftable singularities, the residue vanishes at these points, then we obtain the integral theorem of Cauchy

• Is on holomorphic and has a simple pole with residue in then we obtain the integral formula of Cauchy

Zeros and poles scoring integral

Is on meromorphic with the set of zeros, the pole- and then follows with the residue theorem:

It referred

The zero or Polstellenordnung from to. Applies to the computation rule of the residual for the logarithmic derivative

Practical Application

Using the residue theorem one can compute real integrals with infinite limits of integration. This is carried out in the complex plane, a closed curve, which covers the real limits of integration; the integral over the remaining part of the curve is usually constructed so that it disappears after the border crossing. The complex plane is supplemented by a point at infinity ( Riemann sphere ). This calculation method for improper real integrals is often referred to in theoretical physics as a " method of residuals ."

Cracked Rational Functions

Is the quotient of two polynomials with and for all, is

The upper half- plane, because you can with, for a large, integrated over the closed semi-circle and make the border crossing. Because for large and constant follows the standard estimate for line integrals

So true, and because of the above assessment the latter integral exists also. Using the residue theorem, the formula follows.

Example: Let, with 1st order in Poland. Then, and thus.

Trigonometric Functions

Is the quotient of two polynomials with for all with. Then we have

The unit disk is. Because the number of turns of the unit circle is inside the unit circle, and by assumption there are no singularities on the unit circle.

Example: It is

Because has poles in 1st order, but only the pole at is, and there is the residual.

Fourier transform

Given a function. Furthermore, there is a score, and it should be. Are there then two numbers with large, then for all the formula

The same formula applies to. Using this method, complex Fourier integrals can be calculated. The proof is as above, by cutting the path of integration in the portion on the real axis and the portion in the upper half plane. Thereafter, the threshold is again considered and the integral of the curve in the upper half plane disappears due to lemma Jordan.

The residue theorem for Riemann surfaces

The residue theorem can be generalized to compact Riemann surfaces. A meromorphic 1 form such a surface is that the sum of the residues is equal to zero.

As a consequence, this results in the second set of Liouville on elliptic functions.