Residue (complex analysis)

Summary

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. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points {ak}k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

Definition edit

The residue of a meromorphic function   at an isolated singularity  , often denoted  ,  ,   or  , is the unique value   such that   has an analytic antiderivative in a punctured disk  .

Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a−1 of a Laurent series.

The concept can be used to provide contour integration values of certain contour integral problems considered in the residue theorem. According to the residue theorem, for a meromorphic function  , the residue at point   is given as:

 

where γ is a positively oriented simple closed curve around   and not including any other singularities on or inside the curve.

The definition of a residue can be generalized to arbitrary Riemann surfaces. Suppose   is a 1-form on a Riemann surface. Let   be meromorphic at some point  , so that we may write   in local coordinates as  . Then, the residue of   at   is defined to be the residue of   at the point corresponding to  .

Contour integration edit

Contour integral of a monomial edit

Computing the residue of a monomial

 

makes most residue computations easy to do. Since path integral computations are homotopy invariant, we will let   be the circle with radius   going counter clockwise. Then, using the change of coordinates   we find that

 

hence our integral now reads as

 

Thus, the residue of   is 1 if integer   and 0 otherwise.

Generalization to Laurent series edit

If a function is expressed as a Laurent series expansion around c as follows:

 
Then, the residue at the point c is calculated as:
 
using the results from contour integral of a monomial for counter clockwise contour integral   around a point c. Hence, if a Laurent series representation of a function exists around c, then its residue around c is known by the coefficient of the   term.

Application in Residue theorem edit

For a meromorphic function  , with a finite set of singularities within a positively oriented simple closed curve   which does not pass through any singularity, the value of the contour integral is given according to residue theorem, as:

 
where  , the winding number, is   if   is in the interior of   and   if not, simplifying to:
 
where   are all isolated singularities within the contour  .

Calculation of residues edit

Suppose a punctured disk D = {z : 0 < |zc| < R} in the complex plane is given and f is a holomorphic function defined (at least) on D. The residue Res(f, c) of f at c is the coefficient a−1 of (zc)−1 in the Laurent series expansion of f around c. Various methods exist for calculating this value, and the choice of which method to use depends on the function in question, and on the nature of the singularity.

According to the residue theorem, we have:

 

where γ traces out a circle around c in a counterclockwise manner and does not pass through or contain other singularities within it. We may choose the path γ to be a circle of radius ε around c. Since ε is can be small as we desire it can be made to contain only the singularity of c due to nature of isolated singularities. This may be used for calculation in cases where the integral can be calculated directly, but it is usually the case that residues are used to simplify calculation of integrals, and not the other way around.

Removable singularities edit

If the function f can be continued to a holomorphic function on the whole disk  , then Res(fc) = 0. The converse is not generally true.

Simple poles edit

At a simple pole c, the residue of f is given by:

 

If that limit does not exist, there is an essential singularity there. If it is 0 then it is either analytic there or there is a removable singularity. If it is equal to infinity then the order is higher than 1.

It may be that the function f can be expressed as a quotient of two functions,  , where g and h are holomorphic functions in a neighbourhood of c, with h(c) = 0 and h'(c) ≠ 0. In such a case, L'Hôpital's rule can be used to simplify the above formula to:

 

Limit formula for higher-order poles edit

More generally, if c is a pole of order n, then the residue of f around z = c can be found by the formula:

 

This formula can be very useful in determining the residues for low-order poles. For higher-order poles, the calculations can become unmanageable, and series expansion is usually easier. For essential singularities, no such simple formula exists, and residues must usually be taken directly from series expansions.

Residue at infinity edit

In general, the residue at infinity is defined as:

 

If the following condition is met:

 

then the residue at infinity can be computed using the following formula:

 

If instead

 

then the residue at infinity is

 

For holomorphic functions the sum of the residues at the isolated singularities plus the residue at infinity is zero which gives:

 

Series methods edit

If parts or all of a function can be expanded into a Taylor series or Laurent series, which may be possible if the parts or the whole of the function has a standard series expansion, then calculating the residue is significantly simpler than by other methods. The residue of the function is simply given by the coefficient of   in the Laurent series expansion of the function.

Examples edit

Residue from series expansion edit

Example 1 edit

As an example, consider the contour integral

 

where C is some simple closed curve about 0.

Let us evaluate this integral using a standard convergence result about integration by series. We can substitute the Taylor series for   into the integrand. The integral then becomes

 

Let us bring the 1/z5 factor into the series. The contour integral of the series then writes

 

Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation. The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around C of every other term not in the form cz−1 is zero, and the integral is reduced to

 

The value 1/4! is the residue of ez/z5 at z = 0, and is denoted

 

Example 2 edit

As a second example, consider calculating the residues at the singularities of the function

 
which may be used to calculate certain contour integrals. This function appears to have a singularity at z = 0, but if one factorizes the denominator and thus writes the function as
 
it is apparent that the singularity at z = 0 is a removable singularity and then the residue at z = 0 is therefore 0. The only other singularity is at z = 1. Recall the expression for the Taylor series for a function g(z) about z = a:
 
So, for g(z) = sin z and a = 1 we have
 
and for g(z) = 1/z and a = 1 we have
 
Multiplying those two series and introducing 1/(z − 1) gives us
 
So the residue of f(z) at z = 1 is sin 1.

Example 3 edit

The next example shows that, computing a residue by series expansion, a major role is played by the Lagrange inversion theorem. Let

 
be an entire function, and let
 
with positive radius of convergence, and with  . So   has a local inverse   at 0, and   is meromorphic at 0. Then we have:
 
Indeed,
 
because the first series converges uniformly on any small circle around 0. Using the Lagrange inversion theorem
 
and we get the above expression. For example, if   and also  , then
 
and
 
The first term contributes 1 to the residue, and the second term contributes 2 since it is asymptotic to  .


Note that, with the corresponding stronger symmetric assumptions on   and  , it also follows

 
where   is a local inverse of   at 0.

See also edit

References edit

  • Ahlfors, Lars (1979). Complex Analysis. McGraw Hill.
  • Marsden, Jerrold E.; Hoffman, Michael J. (1998). Basic Complex Analysis (3rd ed.). W. H. Freeman. ISBN 978-0-7167-2877-1.

External links edit