Brewster H.D. Mathematical Physics
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HILARY.
D.
BREWSTER
MATHEMATICAL
PHYSICS
"This page is Intentionally Left Blank"
MATHEMATICAL PHYSICS
Hilary.
D.
Brewster
Oxford Book Company
Jaipur,
India
ISBN: 978-93-80179-02-5
First Edition 2009
Oxford Book Company
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Preface
This book is
intended
to provide
an
account of those parts of
pure
mathematics
that
are
most
frequently
needed
in
physics. This book will
serve several purposes: to provide
an
introduction for
graduate
students
not
previously acquainted
with
the material, to serve as a reference for
mathematical physicists already working in the field,
and
to provide
an
introduction to various
advanced
topics
which
are difficult to
understand
in
the literature.
Not
all
the
techniques
and
application
are treated
in
the
same
depth.
In
general,
we
give a
very
thorough
discussion
of
the
mathematical
techniques
and
applications in
quantum
mechanics,
but
provide only
an
introduction to the
problems
arising
in
quantum
field theory, classical
mechanics,
and
partial differential equations.
This book is for physics
students interested
in
the
mathematics they
use
and
for mathematics
students
interested
in
seeing
how
some of
the
ideas
of
their
discipline
find
realization
in
an
applied
setting.
The
presentation tries to strike a balance
between
formalism
and
application,
between
abstract
and
concrete. The interconnections
among
the various
topics are clarified
both
by
the
use of vector spaces as a central unifying
theme, recurring
throughout
the
book,
and
by
putting
ideas into their
historical context. Enough of the essential formalism is included to make
the
presentation
self-contained. This
book
features
t~
applications of
essential concepts as well as
the
coverage of topics
in
the this field.
Hilary D. Brewster
"This page is Intentionally Left Blank"
Contents
Preface
iii
l.
Mathematical Basics 1
2.
Laplace
and
Saddle Point Method
45
3.
Free Fall
and
Harmonic Oscillators
67
4.
Linear Algebra
107
5.
Complex Representations of Functions
144
6.
Transform
Techniques
in
Physics
191
7.
Problems
in
Higher Dimensions
243
8.
Special Functions
268
Index
288
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Chapter 1
Mathematical
Basics
Before we begin our study
of
mathematical physics, we should review
some mathematical basics. It
is
assumed that you know Calculus and are
comfortable with differentiation and integration.
CALCULUS
IS
IMPORTANT
There are two main topics
in
calculus: derivatives and integrals. You
learned that derivatives are useful
in
providing rates
of
change in either time
or space. Integrals provide areas under curves, but also are useful
in
providing
other types
of
sums over continuous bodies, such as lengths, areas, volumes,
moments
of
inertia, or flux integrals. In physics, one can look at graphs
of
position versus time and the slope (derivative)
of
such a function gives the
velocity.
Then plotting velocity versus time you can either look at the derivative to
obtain acceleration, or you could look at the area under the curve and get the
displacement:
x =
to
vdt.
Of
course, you need to know how to differentiate and integrate given
functions. Even before getting into differentiation and integration, you need
to have a bag
of
functions useful in physics. Common functions are the
polynomial and rational functions. Polynomial functions take the general form
fix)
=
arfXn
+ a
ll
_
1
xn
n
-
1
+
...
+ a
1
x +
ao'
where
an
*-
0.: This
is
the form
of
a polynomial
of
degree
n.
Rational functions
consist
of
ratios
of
polynomials.
Their
graphs
can
exhibit
asymptotes.
Next
are the exponential and
logarithmic functions. The most common are the natural exponential and the
natural logarithm.
The natural exponential is given by
fix)
=~,
where
e::::
2.718281828
....
The natural logarithm is the inverse to the exponential, denoted by
In
x.
The
properties
of
the
expon~tial
function follow from our basic properties for
exponents. Namely, we have: