Week 3
We wish Andrew Bynum to recover well and quickly. Special proposal to Kobe Bryant
for his 61 points at Madison Square Garden.
Outline
Method of partial fractions
–Case 1. The denominator is a product of distinct linear factors
–Case 2. The denominator is a product of linear factors, some of which are repeated
–Case 3. The denominator contains distinct irreducible quadratic factors
–Case 4. The denominator contains a repeated irreducible quadratic factor
How to evaluate coefficients
–Clear the fractions and compare the coefficients (most general argument)
–Heaviside method (often used in case 1)
–Differentiation (often used if denominator has the form (
x

r
)
n
)
Definition of an Improper Integral
–Type I: singularities are infinite
–Type II: singularities are finite
Decide whether an improper integral converges or diverges
–Comparison test
–Limit comparison test
Goal
In the first part our integrands will be the rational functions, which have the form
f
(
x
)
g
(
x
)
,
where
f
and
g
are both polynomials. Recall that the degree of a nonzero polynomial
f
(
x
)
is the largest nonnegative integer
k
such that the coefficient of
x
k
is not zero, denoted by
deg(
g
). For example,
x
4
+ 5
x
3
+ 4
x
+ 5 has degree 4.
Assumption
For the rational function
f
(
x
)
g
(
x
)
, we assume deg(
f
)
<
deg(
g
), otherwise we may apply the
long division to express
f
as
f
(
x
) =
P
(
x
)
g
(
x
)+
R
(
x
), where
P
and
R
are both polynomials
such that deg(
P
) = deg(
f
)

deg(
g
) and deg(
R
)
<
deg(
g
). That is,
f
g
=
P
+
R
g
.
Fundamental Theorem of Algebra
1
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Any polynomial with real coefficients can be written as a product of real linear factors
and real quadratic factors.
That is, if
f
is a polynomial whose coefficient of the term
x
deg(
f
)
is 1, then we can write
f
as
f
(
x
) = (
x

r
1
)
a
1
· · ·
(
x

r
n
)
a
n
(
x
2
+
p
1
x
+
q
1
)
α
1
· · ·
(
x
2
+
p
m
x
+
q
m
)
α
m
,
where
a
i
and
α
j
are all positive integers.
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 Fall '06
 GROSS
 Calculus, Factors, Fractions, Fraction, lim, Rational function

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