I INTRODUCTION, The following is a set of notes to be read in conjunction with the lectures delivered to the 2nd 3rd and. 4th year Students of Physics at Imperial College London. Let us make it clear from the beginning mathematics is the language needed to be able to formulate the. laws of Nature and it is the language needed in order to be able to think about the subtleties of Nature. As with any other language it is difficult to separate the content of a message from the notation and the. syntax Try to formulate the content of Newton s 2nd law without using any mathematical notation. Physicists need mathematics in order to be able to talk and to think The basic mathematics covered. in this course is not only for the theoretical physicists It is a prerequisite for anyone who wants to be. able to read about physics in books or scientific papers or to be able to follow a physics seminar I have. met students who thought they were better physicists because they didn t waist their time or energy on. mathematics This is a misunderstanding Physics as any other reasonably developed science can only. be appreciated by the use of mathematics Not only is mathematics needed to be able to carry out simple. analytic analysis of problems More importantly one is only able to understand the concepts of physics if. one possesses some degree of mathematical flexibility. It is as impossible to learn mathematics solely by watching others perform as it is to learn to play. a musical instrument exclusively by listening to other people play This is not a problem The mind. expanding experience of digesting mathematical concepts through the contemplation of the lecture notes. and the exercises will easily capture the open minded person and lead to a sound engagement with one of. humankind s oldest and most profound activities The musing of mathematical ideas and constructions can. be done anywhere All what is needed is to activate this most marvellous of all instruments our brain And. sometimes pencil and paper comes in handy too In fact it is hardly possible to understand a mathematical. text by reading it It is much better to copy every single detail and to think carefully about what it all. means while coping, The Cambridge book Riley Hobson and Bence Mathematical Methods for Physics and Engineering. together with the Students Solution Manual for Mathematical Methods for Physics and Engineering by Riley. and Hobson is magnificent If one gets hold of this pair of books and read and work with the problems one. will become greatly accomplished in the math used by physics. But there exist many excellent text books on mathematics written for physicists and other scientist I. list a few which I know and have found readable, Introductory level. I1 G Arfken Mathematical Methods for Physicists Academic Press. I2 H Jeffreys and B S Jeffreys Methods of Mathematical Physics Cambridge University Press. I3 D A McQuarrie Mathematical Methods for Scientists and Engineers University Science Books. I4 J Mathews and R L Walker Mathematical Methods of Physics Benjamin Cummings Publishing. Comprehensive more advanced level, A1 Courant and Hilbert Methods of Mathematical Physics. A2 Morse and Feshbach Methods of Theoretical Physics. Here follows a few books that might help one to appreciate the nature of mathematics. E1 J Dieudonne Mathematics The music of reason Springer. E2 G M Phillips Two Millennia of Mathematics Springer. E3 M Aigner and G M Ziegler Proofs from THE BOOK Springer. E4 J R Brown Philosophy of Mathematics Routledge, II COMPLEX FUNCTIONS. Complex functions are not any more complicated than real functions as soon as one has got used to them. That complex numbers are called complex is a matter of history not an indication that they are more. remote and less relevant or less real than real numbers are. That complex numbers are intricately connected to real and natural numbers should be immediately clear. after a few moments of consideration of the famous and spectacular formula. The irrational number denotes the ratio between the circumference and the diameter of a circle The. oldest record of this number is from 1650 BC by an Ahmes Egyptian scibe The irrational number e was. first mentioned in writing in 1618 by Napier in his work on logarithms The natural way to think about e is. to think of a quantity that increases in time with a rate proportional to itself I e a quantity N t think. of the number fragments produced by nuclear fission that evolves according to the equation. It is easy to see that this equation implies an exponential growth of N t Replacing the right hand side by. differences, N t 1 N t 4, and repeating this relation we get. N q 1 q N 0 5, for any natural number q The exact solution to Eq 2 subject to the initial condition N 0 1 is denote. by N t et and hence the number e emerges from the study of rate equations An alternative natural. way of introducing e is through the discussion of logarithms. The imaginary unit was introduced in order to be able to give meaning to roots in equations of the form. x2 1 Imaginary numbers have been refered to ever since Heron of Alexandria 1st century AD but. became only common place after 1500 Today we know that complex numbers are not just a question of. finding roots in mathematical equations they are essential tools for the understanding of the most profound. aspects of reality as captured by Quantum Mechanics. The point we want to make is that Eq 1 relates three numbers from three different ages of human. history in a way that is far from obvious seen in relation to how these numbers were discovered Who. would have expected that the square root of 1 and the ratio between the circumference and the diameter. of a circle are so intimately related Complex numbers and complex functions are as real and as necessary. as the natural numbers, Complex numbers and complex functions are generalisations of their real counter parts They grow out. of the real numbers and functions through the simplest possible generalisations The best way to approach. complex numbers and functions is first to think carefully about definitions and concepts already familiar. from real analysis and then follow the consequences as these definitions are carried over into the complex. regime Differentiation is an important and transparent example of this procedure. FIG 1 The line approaches the tangent as h 0 The slope of the tangent is defined as the derivative. A Differentiation of Complex Functions, Recall how one differentiates a real function The derivative of the function f x at the point x0 can be. interpreted as the slope of the tangent at this point The reason for this is clear Consider the slope of the. line from the point x f x to the point x h f x h See Fig 1 The derivative is obtained as the. value of the slope of this line in the limit h 0 Now note the following important point We only consider. f x to be differentiable if the same value for the slope is obtained irrespectively of how the limit h 0. is assumed We can take the limit while demanding h 0 or we can take the limit while insisting that h. remains non positive The function is only differentiable at x0 if this makes no difference 1. The essential point is that the limit, f x0 x f x df. lim x 0 x 6, is independent of how the limit is taken The derivative of a complex function f z at the point z0 is defined. in exactly the same way, df f z0 z f z, z lim z 0 7. with the same insistence that the limit must not depend on the manner in which the limit is taken This. has remarkably restrictive consequences in the case of complex functions From the requirement that the. same value is obtained for df dz when we let z vary parallel to the x axis as when we let z vary parallel. to the y axis it follows as will be discussed in the lectures that the real part and the imaginary part of. f z are linked together in the following intriguing way. We used the following notation z x iy together with. f z Ref z iImf z, U x y iV x y 9, The relations in Eq 8 are called the Cauchy Riemann conditions Not only does a differentiable function. satisfy these conditions we will find that if a complex function satisfies Eq 8 then the limit in Eq 7. will be independent of how z is taken to zero and hence a function that satisfies Eq 8 will also be. differentiable, Thus to check if a complex function is differentiable one simply check if its real and imaginary parts fulfil. the Cauchy Riemann condition, 1 One can define the derivative from the left or from the right and they might be different if the function has a cusp at x0. but in that case the function is not differentiable only differentiable from the left or the right respectively. FIG 2 Many ways around the mountain but only two different classes. FIG 3 Integration along a path in the complex plane. B Residue theory, Complex differentiable functions posses a spectacular sensitivity to theirs global environment Think of. it this way You are going to cross a big plain from South to North see Fig 2 In the centre there is. a high mountain It is early morning and the sun is in the East You can either pass to the East of the. mountain or to the West Whether you pass two kilometres or three kilometres to the East of the mountain. won t make much difference to you But there is a big difference between passing the mountain to East or. to the West If you decide on the Western route you ll find yourself in the shadow of the mountain for a. while whereas if you stay on the Eastern route you ll be warmed by the sun during the entire trip. It is the same with integrals in the complex plane of complex functions The integral is define much in. the same way as the Riemann sum used to define the integral of a real function Assume that the path P. leads from point zA to point zB in the complex plane Chop the path up in N 1 segments from zA to z1. and from z1 to z2 etc see Fig 3 We introduce dz1 z1 za dz2 z2 z1 dzN zB zN 1 The. following sum, IN f zi dzi 10, is used to define the integral from zA to zB along the path P by assuming the limit N i e we define. f z dz lim IN 11, Consider now the integral along different paths leading from point zA to point zB in the complex plane. If the complex function f z is differentiable everywhere in the plane then as we will see in the lectures. FIG 4 Path not including or including a sigularity. the integral of f z from zA to zB will not depend on which path one follows The situation is however. different if f z has a singularity of the form called a pole If two paths P1 and P2 pass the same way. around this pole then the integrals along the two paths are identical The two integrals are on the other. hand different if the two paths enclose the pole See Fig 4. III FOURIER TRANSFORMATION, We are used to decomposing the 3 dimensional position of a particle r as the sum. r xe1 ye2 ze3 12, along three independent direction vectors e1 e2 and e3 Now think of a function from the real numbers to. say the complex numbers2 f R I C, l Joseph Fourier 1768 1830 discovered that a function f x can be. represented as a sum much in the same way as in Eq 12 The difference is that the sum will typically. include infinitely many terms and that instead of direction vectors we need to use functions It is convenient. to use exponential functions x 7 eikx of different wave vectors k In a schematic fashion we can write. f x f k eikx 13, The important point to pay attention to is that the function f x according to 13 is specified by a set of. coordinates f k one for each value of k and a given set of reference functions. The details are as follows The sum in Eq 13 is for functions defined on the entire real axis replaced. by an integral but that is simply because we may need to include functions eikx for all possible k R. an integral and a sum are in our context essentially the same thing The following two equations express. f x as a superposition of exponentials and give a recipe for how to determine the expansion coefficients. Forward Fourier transformation, f k dxf x e ikx 14. Inverse Fourier transformation, f x f k eikx 15, 2 One might also think of functions from R. I that won t make much difference, The inverse transform can be considered equivalent to the determination of the coefficients in Eq 12. In the case of vectors we determine the coordinates say along the 2nd basis direction e2 from the scalar. product If e2 has length 1 and ei are mutually orthogonal we have y r e2 The integral between f x. and e ikx in Eq 15 is like a scalar product between f x and e ikx. A The power of the Fourier transform, Fourier transformation is a very powerful tool when analysing physical problems We often have to solve. both ordinary as well as partial differential equations In condensed matter physics and statistical mechanics. theoreticians as well as experimentalists use response functions and correlation functions to characterise the. behaviour The mathematical analysis relies on some rather impressive properties of the Fourier transform. E g that differentiation of a function corresponds to a multiplication of its Fourier transform That certain. types of integrals of functions correspond to multiplication of Fourier transforms We list a few of the details. 1 Differential equations, Consider the ordinary differential equation. We know that the solution is obtained from the homogeneous Eq. by adding a particular solution to the inhomogeneous Eq 16 We also kn. together with the Students Solution Manual for Mathematical Methods for Physics and Engineering by Riley and Hobson is magni cent If one gets hold of this pair of books and read and work with the problems one will become greatly accomplished in the math used by physics

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