Bob Tapper Maths MethodsPHYS33010 20089

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PHYS33010 Maths Methods

• Bob Tapper
• room 4.22a e-mail r.j.tapper_at_bris.ac.uk
• http//www.phy.bris.ac.uk/people/tapper_rj/MM3/hom
  e.html
• Lectures Tuesday 1110, Wednesday 1210,
  Thursday 900

AimsTo 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.
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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
• Complex Variables and their Applications, Anthony
  Osborne, Addison-Wesley ISBN 0-201-34290-1
• Schaums Outline Series, Spiegel, McGraw-Hill
  ISBN 0-07-084355-4

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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
  course-materials page for this course ( not on
  Blackboard )

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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)

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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.

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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)
• Here both x and y are real numbers
• is the real part of
• is the imaginary part of

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Introduction

• Addition and Subtraction We define these by
  treating the real and imaginary parts separately
  i.e. the operation

involves adding both parts separately
Given the properties of real numbers we see that
addition and subtraction of complex numbers is
both
Commutative
and
Associative
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Introduction

• Multiplication We define this by using the usual
  rules of algebra to write out all the terms in
  the product

As before, the properties of real numbers mean
that multiplication of complex numbers is both
Commutative
and
Associative
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Introduction

• Reciprocal Almost every complex number
  has a reciprocal which we write as with
  the property that
• Division Multiplication by is
  described as division by

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
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Introduction

• Argand Diagrams Another approach is to regard a
  complex number as an ordered pair of real
  numbers
• Using and as Cartesian coordinates we
  can represent every complex number by a point in
  a plane

y
z
x
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Introduction

• Modulus and Argument Instead of the Cartesian
  coordinates we can use polar coordinates in the
  Argand plane to describe any complex number
• r is called the modulus and q the argument
  of z
• clearly

z
r
q
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Introduction

• Modulus and Argument (cont)
• Clearly
• and
• However care is needed, because many angles have
  the same tangent 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.

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Introduction

• Polar Form The form of a complex number given in
  terms of its modulus and argument
• is called the polar form.
• ( As you know it can be written as
  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
• moduli
  multiply arguments add

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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
• In the polar form the complex conjugate is
  obtained by reversing the sign of the argument.
• Obviously
• and is the length of the vector
  representing
• in the complex plane.

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Introduction

• 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
• by obvious extension

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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
• So
• A similar argument applies to
  and to
• etc etc
• It is often much easier to find by
  using these formulae than to work out
  explicitly and then find its modulus.

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Introduction

• Example if
  find

• Method 1

• Method 2

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Introduction
Definitions of addition and multiplication of
complex quantities allow us to define a huge
number of functions. Example 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.