Methods of Theoretical Physics: I ABSTRACT First-order and second-order differential equations; Wronskian; series solutions; ordi- nary and singular points. Orthogonal eigenfunctions and Sturm-Liouville theory. Complex analysis, contour integration. Integral representations for solutions of ODE’s. Asymptotic expansions. Methods of stationary phase and steepest descent. Generalised functions. Books E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. G. Arfken and H. Weber, Mathematical Methods for Physicists. P.M. Morse and H. Feshbach, Methods of Theoretical Physics. Contents 1 First and Second-order Differential Equations 3 1.1 The Differential Equations of Physics . ..... 3 1.2 First-orderEquations .. .. .. .. .. .. .. .. 4 2 Separation of Variables in Second-order Linear PDE’s 7 2.1 Separation of variables in Cartesian coordinates . ........... 7 2.2 Separation of variables in spherical polar coordinates ............. 10 2.3 Separation of variables in cylindrical polar coordinates............ 12 3 Solutions of the Associated Legendre Equation 13 3.1 Series solution of the Legendre equation . ....... 13 3.2 Properties of the Legendre polynomials . ...... 19 3.3 Azimuthally-symmetric solutions of Laplace’s equation............ 24 3.4 The generating function for the Legendre polynomials . .......... 27 3.5 TheassociatedLegendrefunctions . ..... 29 3.6 The spherical harmonics and Laplace’s equation . ......... 32 3.7 Another look at the generating function . ...... 36 4 General Properties of Second-order ODE’s 39 4.1 Singular points of the equation . 40 4.2 TheWronskian .................................. 42 4.3 Solution of the inhomogeneous equation . ...... 47 4.4 Series solutions of the homogeneous equation . ........ 49 4.5 Sturm-LiouvilleTheory . 65 5 Functions of a Complex Variable 96 5.1 Complex Numbers, Quaternions and Octonions . ...... 96 5.2 Analytic or Holomorphic Functions . 107 5.3 ContourIntegration .............................. 114 5.4 Classification of Singularities . 126 5.5 TheOppenheimFormula ............................ 140 5.6 CalculusofResidues .............................. 144 5.7 Evaluation of real integrals . 145 5.8 SummationofSeries ............................... 156 1 5.9 AnalyticContinuation . 159 5.10 TheGammaFunction .............................. 163 5.11 TheRiemannZetaFunction. 170 5.12 AsymptoticExpansions . 179 5.13 MethodofSteepestDescent . 183 6 Non-linear Differential Equations 189 6.1 MethodofIsoclinals .............................. 190 6.2 Phase-planeDiagrams .. .. .. .. .. .. .. .. 194 7 Cartesian Vectors and Tensors 196 7.1 Rotations and reflections of Cartesian coordinate . .......... 196 7.2 The orthogonal group O(n), and vectors in n dimensions. 199 7.3 Cartesianvectorsandtensors . 201 7.4 Invarianttensors,andthecrossproduct . ....... 205 7.5 CartesianTensorCalculus . 213 2 1 First and Second-order Differential Equations 1.1 The Differential Equations of Physics It is a phenomenological fact that most of the fundamental equations that arise in physics are of second order in derivatives. These may be spatial derivatives, or time derivatives in various circumstances. We call the spatial coordinates and time, the independent variables of the differential equation, while the fields whose behaviour is governed by the equation are called the dependent variables. Examples of dependent variables are the electromag- netic potentials in Maxwell’s equations, or the wave function in quantum mechanics. It is frequently the case that the equations are linear in the dependent variables. Consider, for example, the scalar potential φ in electrostatics, which satisfies 2φ = 4π ρ (1.1) ∇ − where ρ is the charge density. The potential φ appears only linearly in this equation, which is known as Poisson’s equation. In the case where there are no charges present, so that the right-hand side vanishes, we have the special case of Laplace’s equation. Other linear equations are the Helmholtz equation 2ψ+k2 ψ = 0, the diffusion equation ∇ 2ψ ∂ψ/∂t = 0, the wave equation 2ψ c−2 ∂2ψ/∂t2 = 0, and the Schr¨odinger equation ∇ − ∇ − ¯h2/(2m) 2ψ + V ψ i¯h∂ψ/∂t = 0. − ∇ − The reason for the linearity of most of the fundamental equations in physics can be traced back to the fact that the fields in the equations do not usually act as sources for themselves. Thus, for example, in electromagnetism the electric and magnetic fields respond to the sources that create them, but they do not themselves act as sources; the electromagnetic fields themselves are uncharged; it is the electrons and other particles that carry charges that act as the sources, while the photon itself is neutral. There are in fact generalisations of Maxwell’s theory, known as Yang-Mills theories, which play a fundamental rˆole in the description of the strong and weak nuclear forces, which are non-linear. This is precisely because the Yang-Mills fields themselves carry the generalised type of electric charge. Another fundamental theory that has non-linear equations of motion is gravity, described by Einstein’s general theory of relativity. The reason here is very similar; all forms of energy (mass) act as sources for the gravitational field. In particular, the energy in the gravitational field itself acts as a source for gravity, hence the non-linearity. Of course in the Newtonian limit the gravitational field is assumed to be very weak, and all the non-linearities disappear. In fact there is every reason to believe that if one looks in sufficient detail then even the linear Maxwell equations will receive higher-order non-linear modifications. Our best 3 candidate for a unified theory of all the fundamental interactions is string theory, and the way in which Maxwell’s equations emerge there is as a sort of “low-energy” effective theory, which will receive higher-order non-linear corrections. However, at low energy scales, these terms will be insignificantly small, and so we won’t usually go wrong by assuming that Maxwell’s equations are good enough. The story with the order of the fundamental differential equations of physics is rather similar too. Maxwell’s equations, the Schr¨odinger equation, and Einstein’s equations are all of second order in derivatives with respect to (at least some of) the independent variables. If you probe more closely in string theory, you find that Maxwell’s equations and the Einstein equations will also receive higher-order corrections that involve larger numbers of time and space derivatives, but again, these are insignificant at low energies. So in some sense one should probably ultimately take the view that the fundamental equations of physics tend to be of second order in derivatives because those are the only important terms at the energy scales that we normally probe. We should certainly expect that at least second derivatives will be observable, since these are needed in order to describe wave-like motion. For Maxwell’s theory the existence of wave-like solutions (radio waves, light, etc.) is a commonplace observation, and probably in the not too distant future gravitational waves will be observed too. 1.2 First-order Equations Differential equations involving only one independent variable are called ordinary differen- tials equations, or ODE’s, by contrast with partial differential equations, or PDE’s, which have more than one independent variable. Even first-order ODE’s can be complicated. One situation that is easily solvable is the following. Suppose we have the single first- order ODE dy(x) = F (x) . (1.2) dx The solution is, of course, simply given by y(x) = x dx′F (x′) (note that x′ here is just a name for the “dummy” integration variable). ThisR is known as “reducing the problem to quadratures,” meaning that it now comes down to just performing an indefinite integral. Of course it may or may not be be that the integral can be evaluated explicitly, but that is a different issue; the equation can be regarded as having been solved. The most general first-order ODE takes the form dy(x) = F (x,y) . (1.3) dx 4 A special class of function F (x,y) for which we can again easily solve the equation explicitly is if P (x) F (x,y)= , (1.4) − Q(y) implying that (1.3) becomes P (x) dx + Q(y) dy = 0, since then we can reduce the solution to quadratures, with x y dx′ P (x′)+ dy′ Q(y′) = 0 . (1.5) Z Z Note that no assumption of linearity is needed here. A rather more general situation is when P (x,y) F (x,y)= , (1.6) −Q(x,y) and if the differential P (x,y) dx + Q(x,y) dy is exact, which means that we can find a function ϕ(x,y) such that dϕ(x,y)= P (x,y) dx + Q(x,y) dy . (1.7) Of course there is no guarantee that such a ϕ will exist. Clearly a necessary condition is that ∂P (x,y) ∂Q(x,y) = , (1.8) ∂y ∂x since dϕ = ∂ϕ/∂x dx + ∂ϕ/∂y dy, which implies we must have ∂ϕ ∂ϕ = P (x,y) , = Q(x,y) , (1.9) ∂x ∂y since second partial derivatives of ϕ commute: ∂2ϕ ∂2ϕ = . (1.10) ∂x∂y ∂y∂x In fact, one can also see that (1.8) is sufficient for the existence of the function ϕ; the condition (1.8) is known as an integrability condition for ϕ to exist. If ϕ exists, then solving the differential equation (1.3) reduces to solving dϕ = 0, implying ϕ(x,y) = c =constant. Once ϕ(x,y) is known, this implicitly gives y as a function of x. If P (x,y) and Q(x,y) do not satisfy (1.8) then all is not lost, because we can recall that solving the differential equation (1.3), where F (x,y) = P (x,y)/Q(x,y) means solving − P (x,y) dx + Q(x,y) dy = 0, which is equivalent to solving α(x,y) P (x,y) dx + α(x,y) Q(x,y) dy = 0 , (1.11) 5 where α(x,y) is some generically non-vanishing but as yet otherwise arbitrary function.