Partial fractions in complex analysis

In complex analysis, a partial fraction expansion is a way of writing a meromorphic function as an infinite sum of rational functions and polynomials. When is a rational function, this reduces to the usual method of partial fractions.

Motivation

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By using polynomial long division and the partial fraction technique from algebra, any rational function can be written as a sum of terms of the form  , where   and   are complex,   is an integer, and   is a polynomial. Just as polynomial factorization can be generalized to the Weierstrass factorization theorem, there is an analogy to partial fraction expansions for certain meromorphic functions.

A proper rational function (one for which the degree of the denominator is greater than the degree of the numerator) has a partial fraction expansion with no polynomial terms. Similarly, a meromorphic function   for which   goes to 0 as   goes to infinity at least as quickly as   has an expansion with no polynomial terms.

Calculation

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Let   be a function meromorphic in the finite complex plane with poles at   and let   be a sequence of simple closed curves such that:

  • The origin lies inside each curve  
  • No curve passes through a pole of  
  •   lies inside   for all  
  •  , where   gives the distance from the curve to the origin
  • one more condition of compatibility with the poles  , described at the end of this section

Suppose also that there exists an integer   such that

 

Writing   for the principal part of the Laurent expansion of   about the point  , we have

 

if  . If  , then

 

where the coefficients   are given by

 

  should be set to 0, because even if   itself does not have a pole at 0, the residues of   at   must still be included in the sum.

Note that in the case of  , we can use the Laurent expansion of   about the origin to get

 
 
 

so that the polynomial terms contributed are exactly the regular part of the Laurent series up to  .

For the other poles   where  ,   can be pulled out of the residue calculations:

 
 
  • To avoid issues with convergence, the poles should be ordered so that if   is inside  , then   is also inside   for all  .

Example

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The simplest meromorphic functions with an infinite number of poles are the non-entire trigonometric functions. As an example,   is meromorphic with poles at  ,   The contours   will be squares with vertices at   traversed counterclockwise,  , which are easily seen to satisfy the necessary conditions.

On the horizontal sides of  ,

 

so

 
 

  for all real  , which yields

 

For  ,   is continuous, decreasing, and bounded below by 1, so it follows that on the horizontal sides of  ,  . Similarly, it can be shown that   on the vertical sides of  .

With this bound on   we can see that

 

That is, the maximum of   on   occurs at the minimum of  , which is  .

Therefore  , and the partial fraction expansion of   looks like

 

The principal parts and residues are easy enough to calculate, as all the poles of   are simple and have residue -1:

 
 

We can ignore  , since both   and   are analytic at 0, so there is no contribution to the sum, and ordering the poles   so that  , etc., gives

 
 

Applications

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Infinite products

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Because the partial fraction expansion often yields sums of  , it can be useful in finding a way to write a function as an infinite product; integrating both sides gives a sum of logarithms, and exponentiating gives the desired product:

 
 
 

Applying some logarithm rules,

 
 

which finally gives

 

Laurent series

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The partial fraction expansion for a function can also be used to find a Laurent series for it by simply replacing the rational functions in the sum with their Laurent series, which are often not difficult to write in closed form. This can also lead to interesting identities if a Laurent series is already known.

Recall that

 

We can expand the summand using a geometric series:

 

Substituting back,

 

which shows that the coefficients   in the Laurent (Taylor) series of   about   are

 
 

where   are the tangent numbers.

Conversely, we can compare this formula to the Taylor expansion for   about   to calculate the infinite sums:

 
 
 

See also

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References

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  • Markushevich, A.I. Theory of functions of a complex variable. Trans. Richard A. Silverman. Vol. 2. Englewood Cliffs, N.J.: Prentice-Hall, 1965.