How do you compute something like
So far you have no method for doing this.
The trick (which is called partial fraction decomposition),
is to write
The integral on the right is then
easy to do (the answer involves 's).
But how on earth do you right the rational function
on the left hand side as a sum of the nice terms
of the right hand side? Doing this is called
``partial fraction decomposition'', and it is
a fundamental idea in mathematics. It relies on
our ability to factor polynomials and saolve linear
equations. As a first hint, notice that
so the denominators in the decomposition correspond to
the factors of the denominator.
Before describing the secret behind (5.4.1), we'll
discuss some background about how polynomials and rational functions
(Fundamental Theorem of Algebra)
is a polynomial,
then there are complex numbers
If is a polynomial, the roots of
correspond to the factors of . Thus if
for each (and nowhere else).
For example, if
then is a zero with multiplicity ,
is a zero with multiplicity , and is a ``zero
(Multiplicity of Zero)
The multiplicity of a zero
is the number
of times that
appears as a factor of
A rational function
is a quotient
is a rational function.
of a rational function
is a complex number
is unbounded as
For example, for (5.4.2) the poles
are at , , and . They have
multiplicity , , and , respectively.