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Bob Tapper – Maths MethodsPHYS /9 PHYS33010 Maths Methods Bob Tapper room: 4.22a 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.

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Bob Tapper – Maths MethodsPHYS /9 Recommended Texts General Mathematics books: Mathematical Methods in the Physical Sciences, Mary Boas, John Wiley & Sons ISBN (~£35) Mathematical Methods for Physicists, Arfken and Weber, Academic Press ISBN More specialised books: Complex Variables and their Applications, Anthony Osborne, Addison-Wesley ISBN Schaums Outline Series, Spiegel,McGraw-Hill ISBN

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Bob Tapper – Maths MethodsPHYS /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 course- materials page for this course ( not on Blackboard )

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Bob Tapper – Maths MethodsPHYS /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)

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Bob Tapper – Maths MethodsPHYS /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.

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Bob Tapper – Maths MethodsPHYS /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): Here both x and y are real numbers: is the real part of is the imaginary part of

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Bob Tapper – Maths MethodsPHYS /9 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 Commutativeand Associative

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Bob Tapper – Maths MethodsPHYS /9 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 Commutativeand Associative

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Bob Tapper – Maths MethodsPHYS /9 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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Bob Tapper – Maths MethodsPHYS /9 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: z x y

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Bob Tapper – Maths MethodsPHYS /9 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 the argument of z clearly z r

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Bob Tapper – Maths MethodsPHYS /9 Introduction Modulus and Argument (cont): Clearly and However care is needed, because many angles have the same tangent so 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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Bob Tapper – Maths MethodsPHYS /9 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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Bob Tapper – Maths MethodsPHYS /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 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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Bob Tapper – Maths MethodsPHYS /9 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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Bob Tapper – Maths MethodsPHYS /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: 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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Bob Tapper – Maths MethodsPHYS /9 Introduction Example: if find Method 1: Method 2:

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Bob Tapper – Maths MethodsPHYS /9 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.

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