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Asymptotics and Special Functions
Frank Olver
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eBook - PDF
Asymptotics and Special Functions
Frank Olver
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About This Book
A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.
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Information
INTRODUCTION
TO
ASYMPTOTIC
ANALYSIS
1
Origin
of
Asymptotic
Expansions
1.1
Consider
the
integral
F(x)
=
re-'
cost
dr
for
positive
real
values
of
the
parameter
x.
Let
us
attempt
its
evaluation
by
expanding
cost
in
powers
of
t
and
integrating
the
resulting
series
term
by
term.
We
obtain
Provided
that
x
>
1
the
last
series
converges
to
the
sum
That
the
attempt
proved
to
be
successful
can
be
confirmed
by
deriving
the
last
result
directly
from
(1.01)
by
means
of
two
integrations
by
parts;
the
restriction
x
>
1
is
then
seen
to
be
replaceable
by
x
>
0.
Now
let
us
follow
the
same
procedure
with
the
integral
We
obtain