Residual (function theory)

from Wikipedia, the free encyclopedia

In function theory , the residual of a complex-valued function is an aid for calculating complex curve integrals using the residual theorem .

definition

Complex areas

Be one area , isolated in and holomorphic . Then for every point there exists a dotted neighborhood that is relatively compact in , with holomorphic. In this case, has to a Laurent development . Then one defines for the residual of in

.

Riemann number ball

The above definition can also be extended to the Riemann number sphere . Again let be a discrete set in and a holomorphic function. Then for all with the residual is also explained by the above definition. For one sets

where is a circle with a sufficiently large radius that is oriented clockwise, and is the -1 as above. Laurent series coefficient.

Features and Notes

  • Let be a domain and a holomorphic function in . Then Cauchy's integral theorem can be applied, from which it follows that the residual of in is zero.
  • The integral representation shows that one can also speak of the residual of the differential form .
  • The residual theorem applies .
  • For rational functions , the so-called unity relation holds: . Here is the set of all poles of and the Riemann number sphere .

Practical calculation

The following rules can be used in practice to calculate residuals of complex-valued functions in point :

  • The residual is -linear, i.e. H. for the following applies:
  • Has in a pole first order, the following applies:
  • Has in a pole first order and is in holomorphic, then:
  • Has in a zero first order, the following applies:
  • Has in a zero first order and is in holomorphic, then:
  • Has in a pole -th order, the following applies:
  • Has in a zero th order, then: .
  • Has in a zero th order and g in holomorphic, then: .
  • Has in a pole th order, then: .
  • Has in a pole th order and g in holomorphic, then: .
  • If the residual at the point is to be calculated, then applies . Because with applies

The rules about the logarithmic derivative are also of theoretical interest in connection with the residual theorem.

Examples

  • As mentioned earlier, when is on an open environment of holomorph.
  • If so has in a pole first order, and it is .
  • As you can see immediately with the linearity and the rule of the logarithmic derivative, because it has a first order zero.
  • The continued gamma function has in for poles of 1st order, and the residual there is .

Algebraic view

Let there be a body and a connected regular actual curve over . Then there is a canonical map for every closed point

which assigns its residual in to every meromorphic differential form .

If a -rational point and a local uniformizer, then the residual mapping can be specified explicitly as follows: Is a meromorphic differential form and a local representation, and is

the Laurent series of , then applies

In particular, the algebraic residual in the case agrees with the function-theoretical one.

The analog of the residual theorem is correct: For every meromorphic differential form the sum of the residuals is zero:

swell

A construction of the algebraic residual mapping.