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Proof.
See [Buck78].
Definition.
The integrals in (D.1) are called
Fresnel integrals
.
D.2
Series
This section reviews some basic facts about series, in particular Taylor polynomials
and series.
Definition.
A series
•
Â
0
a
n
(D.2)
n =
is said to
converge
to the sum A if the sequence of partial sums
k
Â
0
a
n
n
=
converges to A as k goes to infinity, otherwise, it is said to
diverge
. If the series (D.2)
converges, but the series
•
Â
0
a
n
(D.3)
n=
diverges, (D.2) is called a
conditionally convergent
series. If (D.3) converges, then (D.2)
is called an
absolutely convergent
series.
D.2.1. Theorem.
Every absolutely convergent series converges.
Proof.
See [Buck78].
Definition.
Series of the form
•
Â
0
n
ax
(D.4)
n
n =
or
•
Â
0
n
axc
n
(
-
)
(D.5)
n
=
are called
power series
in x or x - c, respectively.
D.2.2. Theorem.
For every power series of the form (D.4) there is an R, 0 £ R £•,
so that the series converges absolutely for all x, |x| < R, and diverges for all x, R < |x|.
One can compute R with the formulas
