# Evaluation Integral: Method of Partial Fractions - Can the system of equations always be solved?

I've been studying the method of partial fractions for evaluation integrals. So far every example and exercise have been fairly straight forward, but I still have some unanswered questions:

1) The standard table for doing method of partial fractions - I guess some clever person has come up with this table and verified it works. However is this the only way of doing partial fractions, or are there other tables / methods around ?

2) The system of equations to be solved from using method of partial fractions - Does this system always has a solution ? So far I've never experienced a system with no solutions, but that is probably because classroom exercises are solvable. Can one come up with a system of equations not solvable for constants $A$, $B$, ... $K$, such that method of partial fractions fails ?

Thanks for your time.

## 2 Answers

In calculus course, and in engineering-math courses doing Lapace transforms, the method of partial fractions is typically stated without the proof that the equations can always be solved. After all, once you solve the system of equations, you can check your answer without needing that theory.

Such a proof could be done later in a linear algebra course (showing a certain determinant does not vanish), but I do not recall seeing that in a modern linear algebra textbook. So maybe look in textbooks from before 1950.

Such a proof can also be done in a complex analysis textbook: in that case (using complex numbers) we do not need quadratic factors in our denominator, and things work out well. This may be in a chapter on Laurent series. You find the principal part at each pole (including infinity), then after you subtract those, you have a bounded entire function, hence a constant.

• Thanks for your answer. Could you tell me if the constants are unique in the partial fraction decomposition ? – Shuzheng Nov 30 '13 at 9:50
• The constants are unique. – GEdgar Nov 30 '13 at 16:47
• Thanks, this can be proved using that polynomial division is unique in a field ? Like stated here mathforum.org/library/drmath/view/51687.html etc ? – Shuzheng Nov 30 '13 at 17:51

It might be worth taking a look at a (fairly old) book by G. H. Hardy on Integration of Functions of a Single Variable. The book is freely available on Project Gutenberg

On pages 11 onwards he discusses rational functions and partial fractions and states:

"the integral of any rational function is an elementary function
which is rational save for the possible presence of logarithms of
rational functions"


Basically he is deducing this from the fundamental theorem of algebra.