PHYS33010 Maths Methods
Bob Tapper
room: 4.22a e-mail: [email protected]
http://www.phy.bris.ac.uk/people/tapper_rj/MM3/home.html
Lectures: Tuesday 11:10, Wednesday 12:10, Thursday 9:00
Aims:To introduce a range of powerful mathematical techniques
for solving physics problems. These methods include complex
variable theory, Fourier and Laplace transforms. With these
tools, to show how to solve many of the differential equations
arising in different branches of physics.
Bob Tapper – Maths Methods
PHYS33010 2008/9
Recommended Texts

General Mathematics books:
Mathematical Methods in the Physical Sciences , Mary
Boas, John Wiley & Sons ISBN 0-471-04409-1 (~£35)
 Mathematical Methods for Physicists, Arfken and
Weber, Academic Press ISBN 0-12-059816-7
More specialised books:

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
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Complex Variables and their Applications, Anthony
Osborne, Addison-Wesley ISBN 0-201-34290-1
Schaum’s Outline Series, Spiegel, McGraw-Hill ISBN
0-07-084355-4
Bob Tapper – Maths Methods
PHYS33010 2008/9
Presentation

The Introduction (which is mainly revision) will be
delivered using Powerpoint

Rest of the course will be written out on a document
projector with occasional handouts

We will illustrate the use of Maple to solve some of the
problems we encounter, and especially to generate
beautiful graphics illustrating the behaviour of functions
etc.

Links to all the material will be placed on the “coursematerials” page for this course ( not on Blackboard )
Bob Tapper – Maths Methods
PHYS33010 2008/9
Syllabus

Introduction: Complex numbers (mostly revision).

Functions of a complex variable.

Contour integrals. Finding real integrals by contour
integration

Fourier Transforms (contd) and Laplace Transforms

Using integral transforms to solve Differential Equations

Further Applications. Conformal mapping (if we have time)
Bob Tapper – Maths Methods
PHYS33010 2008/9
Problem Sheets and Classes



It is absolutely essential for you to do numerous problems.
Only in this way can you develop a full understanding of
the material.
There are three scheduled problems classes for this
course.In these you will do some unseen exercises to
which answers will be provided during the class. Both
problems and solutions will be placed on the web after the
class.
In addition there will be four problem sheets which will be
distributed during the course. Solutions to these will be
provided on the web a week or so later.
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction
Complex Numbers: As you all know, complex numbers are obtained by
extending the familiar real numbers by using the symbol “i” (or in
engineering “j”) to represent the square root of –1 (which is clearly
not a real number):
z  x  iy
Here both x and y are real numbers:
x
y
is the “real part” of
z
is the “imaginary part” of
Bob Tapper – Maths Methods
z
PHYS33010 2008/9
Introduction

Addition and Subtraction: We define these by treating the
real and imaginary parts separately: i.e. the operation
z  z1  z2
involves adding both parts separately:
z  z1  z2  x1  iy1  x2  iy2
 ( x1  x2 )  i ( y1  y2 )
Given the properties of real numbers we see that addition
and subtraction of complex numbers is both
Commutative
and
1
2
2
1
z z z z
Associative
z1  ( z2  z3 )  ( z1  z2 )  z3  z1  z2  z3
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction

Multiplication: We define this by using the usual rules of
algebra to write out all the terms in the product:
z1 z2  ( x1  iy1 )( x2  iy2 )
2
 x1 x2  i ( x1 y2  y1 x2 )  i y1 y2
 x1 x2  y1 y2  i ( x1 y2  y1 x2 )
As before, the properties of real numbers mean that
multiplication of complex numbers is both
Commutative
and
1 2
2 1
zz z z
Associative
z1 ( z2 z3 )  ( z1 z2 ) z3  z1 z2 z3
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction

z
Reciprocal: Almost every complex number
has a
1
reciprocal which we write as
with the property that:
z
zz  1
1

Division: Multiplication by
by
z
z
1
is described as division
All these properties are the same as those of real
numbers so we can carry over all the usual rules for
manipulating algebraic expressions to the situation
where the symbols represent complex numbers
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction

Argand Diagrams: Another approach is to regard a complex
number as an ordered pair of real numbers:
z  ( x, y )

Using x and y as Cartesian coordinates we can
represent every complex number by a point in a plane:
y
z
x
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction

Modulus and Argument: Instead of the Cartesian
coordinates we can use polar coordinates in the Argand
plane to describe any complex number:
z
r
q


r is called the “modulus” and q the “argument” of z
clearly
Bob Tapper – Maths Methods
x  Re( z )  r cosq
y  Im( z )  r sin q
PHYS33010 2008/9
Introduction
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Modulus and Argument (cont):

Clearly

and

However care is needed, because
many angles have the same tangent
z  x  iy  r (cos q  i sin q )
r | z | x 2  y 2
1
q  arg( z )  tan ( y / x)
so q is multivalued. We define the
“principal value” of the argument by
measuring it anticlockwise from the
positive-going real axis in the
complex plane.
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction

Polar Form: The form of a complex number given in terms
of its modulus and argument:
z  r cosq  ir sin q
is called the “polar form”.
( As you know it can be written as z  re iq but we have
not yet defined the meaning of the right-hand side)
Modulus and argument behave in a simpler way than real
and imaginary parts when complex numbers are
multiplied or divided:
If
z  z1 z2
r  r1r2
“moduli multiply”
Bob Tapper – Maths Methods
q  q1  q 2
“arguments add”
PHYS33010 2008/9
Introduction

Complex Conjugate: An important operation on a complex
number is the formation of its complex conjugate, given
by reversal of the sign of its imaginary part:
If
z *  z  x  iy
z  x  iy
In the “polar form” the complex conjugate is obtained by
reversing the sign of the argument.
Obviously
and
|z|
zz  z z  x  y  r  | z |
*
*
2
2
2
is the length of the vector representing
2
z
in the complex plane.
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction
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Triangle Inequality: Addition of two complex numbers is a
simple vector sum in an Argand diagram.
from the geometrical properties of the triangle we see that
| z1  z2 |  | z1 |  | z2 |
by obvious extension
| z1  z2  z3 |  | z1 |  | z2 |  | z3 |
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction

Modulus of Product: If a complex number is given as the
product of two other complex numbers
it is
simple to find its modulus:
z  uv
| z |  z z  (uv) (uv)  u uv v | u | | v |
| z || u || v |
2



So
*
*
A similar argument applies to
z  uv / w
*
z  u/v
2
2
and to
etc etc
It is often much easier to find
formulae than to work out z
modulus.
Bob Tapper – Maths Methods
*
|z|
by using these
explicitly and then find its
PHYS33010 2008/9
Introduction
if z  a  ib
find | z |2
c  id
| z |2 | a  ib |2 / | c  id |2  (a 2  b 2 ) /(c 2  d 2 )

Example:

Method 1:

Method 2: z  (a  ib)(c  id )  ac  bd  i bc  ad
2
2
2
2
(c  id )(c  id ) c  d
c d
(ac  bd ) 2  (bc  ad ) 2
2
2
2
| z |  Re( z )  Im( z ) 
(c 2  d 2 ) 2
(a 2 c 2  2abcd  b 2 d 2 )  (b 2 c 2  2abcd  a 2 d 2 )

(c 2  d 2 ) 2
2 2
2
2
2 2
2
2
a 2 (c 2  d 2 )  b 2 (c 2  d 2 ) a 2  b 2
a
c

b
d

b
c

a
d



2
2 2
2
2 2
(c  d )
(c  d )
c2  d 2
Bob Tapper – Maths Methods
PHYS33010 2008/9
Introduction
Definitions of addition and multiplication of complex quantities
allow us to define a huge number of functions.
Example:
z 6  7 z ( z  1)
f z  
z2  z  3
However, we need to be able to define and use an even wider
class of functions, such as sines and logarithms. One way this
can be done in real analysis is by the use of infinite series, and
this method can also be used for complex analysis.
Bob Tapper – Maths Methods
PHYS33010 2008/9