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©2009 by L. Lagerstrom
The Partial Fraction Expansion
• For an introduction to the basic concepts, see the
associated video clip
• Motivation via the Laplace Transform
• Proper and improper rational functions
• Basic form of the partial fraction expansion (PFE)
• Finding the residues and roots
• Using the residue function
• Repeated roots and the PFE
• Improper rational functions and the PFE
©2009 by L. Lagerstrom
Motivation via the Laplace Transform
In certain types of engineering analysis, such as the analysis of
circuits, a mathematical technique involving something called the
Laplace transform is very helpful. The details of the Laplace
transform are beyond this course, but the results that sometimes
occur when it's used are important for our current topic (the "partial
fraction expansion").
In particular: When using the Laplace transform to analyze a circuit,
we get an integral of the following form:
where A(x) and B(x) are polynomials. In general, this is not an easy
integral to solve. But by using this "partial fraction expansion"
technique, we can solve it relatively easily.
∫
dx
x
B
x
A
)
(
)
(
©2009 by L. Lagerstrom
Proper and Improper Rational Functions
Before we introduce the partial fraction expansion (PFE), we need
the concepts of a "proper rational function" and an "improper rational
function".
Consider the ratio of two polynomials, A(x) and B(x):
We say that H(x) is a proper rational function if the order (degree) of
the numerator polynomial, A(x), is less than the order of the
denominator polynomial, B(x). So, for example, if A(x) is a quadratic
(secondorder) polynomial and B(x) is a cubic (thirdorder)
polynomial, then H(x) is proper.
If the order of A(x) is greater than or equal to the order of B(x), then
we say that H(x) is an improper rational function.
This matters because the partial fraction expansion only works if
H(x) is a proper rational function.
)
(
)
(
)
(
x
B
x
A
x
H
=
©2009 by L. Lagerstrom
Basic Form of the PFE
Let's consider a proper rational function H(x) = A(x)/B(x). Although we
will leave the proof to your math class, it turns out that it's possible to
expand a proper rational function as a sum of terms thusly:
In this expression the c's are constants known as "residues" and the r's
are the roots of the denominator polynomial B(x). (The variable x is of
course the independent variable.)
We note that if we use the PFE to replace the ratio A(x)/B(x) in the
Laplace transform integral, it becomes easy to solve, because it is now
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 Spring '10
 Lagerstrom
 Fraction, Rational function, Partial fractions in integration, L. Lagerstrom, residue function

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