Mathematical Methods for Physics Peter S. Riseborough June 18, 2018 Contents 1 Mathematics and Physics 5 2 Vector Analysis 6 2.1 Vectors . 6 2.2 Scalar Products . 7 2.3 The Gradient . 7 2.4 The Divergence . 8 2.5 The Curl . 10 2.6 Successive Applications of r .................... 12 2.7 Gauss's Theorem . 13 2.8 Stokes's Theorem . 15 2.9 Non-Orthogonal Coordinate Systems . 16 2.9.1 Curvilinear Coordinate Systems . 18 2.9.2 Spherical Polar Coordinates . 19 2.9.3 The Gradient . 21 2.9.4 The Divergence . 21 2.9.5 The Curl . 22 2.9.6 Compounding Vector Differential Operators in Curvilin- ear Coordinates . 23 3 Partial Differential Equations 27 3.1 Linear First-Order Partial Differential Equations . 31 3.2 Classification of Partial Differential Equations . 37 3.3 Boundary Conditions . 48 3.4 Separation of Variables . 48 4 Ordinary Differential Equations 64 4.1 Linear Ordinary Differential Equations . 66 4.1.1 Singular Points . 67 4.2 The Frobenius Method . 68 4.2.1 Ordinary Points . 69 1 4.2.2 Regular Singularities . 74 4.3 Linear Dependence . 86 4.3.1 Linearly Independent Solutions . 90 4.3.2 Abel's Theorem . 90 4.3.3 Other Solutions . 91 5 St¨urmLiouville Theory 93 5.1 Degenerate Eigenfunctions . 98 5.2 The Inner Product . 99 5.3 Orthogonality of Eigenfunctions . 100 5.4 Orthogonality and Linear Independence . 102 5.5 Gram-Schmidt Orthogonalization . 102 5.6 Completeness of Eigenfunctions . 106 6 Fourier Transforms 112 6.1 Fourier Transform of Derivatives . 113 6.2 Convolution Theorem . 116 6.3 Parseval's Relation . 117 7 Fourier Series 119 7.1 Gibbs Phenomenon . 123 8 Bessel Functions 133 8.0.1 The Generating Function Expansion . 133 8.0.2 Series Expansion . 134 8.0.3 Recursion Relations . 134 8.0.4 Bessel's Equation . 136 8.0.5 Integral Representation . 137 8.0.6 Addition Theorem . 138 8.0.7 Orthonormality . 144 8.0.8 Bessel Series . 146 8.1 Neumann Functions . 153 8.2 Spherical Bessel Functions . 157 8.2.1 Recursion Relations . 158 8.2.2 Orthogonality Relations . 159 8.2.3 Spherical Neumann Functions . 159 9 Legendre Polynomials 162 9.0.4 Generating Function Expansion . 162 9.0.5 Series Expansion . 163 9.0.6 Recursion Relations . 163 9.0.7 Legendre's Equation . 168 9.0.8 Orthogonality . 169 9.0.9 Legendre Expansions . 171 2 9.1 Associated Legendre Functions . 181 9.1.1 The Associated Legendre Equation . 181 9.1.2 Generating Function Expansion . 184 9.1.3 Recursion Relations . 185 9.1.4 Orthogonality . 188 9.2 Spherical Harmonics . 192 9.2.1 Expansion in Spherical Harmonics . 192 9.2.2 Addition Theorem . 193 10 Hermite Polynomials 197 10.0.3 Recursion Relations . 197 10.0.4 Hermite's Differential Equation . 197 10.0.5 Orthogonality . 198 11 Laguerre Polynomials 201 11.0.6 Recursion Relations . 201 11.0.7 Laguerre's Differential Equation . 202 11.1 Associated Laguerre Polynomials . 202 11.1.1 Generating Function Expansion . 202 12 Inhomogeneous Equations 208 12.1 Inhomogeneous Differential Equations . 208 12.1.1 Eigenfunction Expansion . 209 12.1.2 Piece-wise Continuous Solution . 209 12.2 Inhomogeneous Partial Differential Equations . 218 12.2.1 The Symmetry of the Green's Function. 219 12.2.2 Eigenfunction Expansion . 230 13 Complex Analysis 242 13.1 Contour Integration . 245 13.2 Cauchy's Integral Theorem . 248 13.3 Cauchy's Integral Formula . 254 13.4 Derivatives . 256 13.5 Morera's Theorem. 258 14 Complex Functions 261 14.1 Taylor Series . 261 14.2 Analytic Continuation . 262 14.3 Laurent Series . 265 14.4 Branch Points and Branch Cuts . 267 14.5 Singularities . 271 3 15 Calculus of Residues 275 15.1 Residue Theorem . 275 15.2 Jordan's Lemma . 277 15.3 Cauchy's Principal Value . 278 15.4 Contour Integration . 282 15.5 The Poisson Summation Formula . 294 15.6 Kramers-Kronig Relations . 298 15.7 Integral Representations . 300 4 1 Mathematics and Physics Physics is a science which relates measurements and measurable quantities to a few fundamental laws or principles. It is a quantitative science, and as such the relationships are mathematical. The laws or principles of physics must be able to be formulated as mathematical statements. If physical laws are to be fundamental, they must be few in number and must be able to be stated in ways which are independent of any arbitrary choices. In particular, a physical law must be able to be stated in a way which is in- dependent of the choice of reference frame in which the measurements are made. The laws or principles of physics are usually formulated as differential equa- tions, as they relate changes. The laws must be invariant under the choice of coordinate system. Therefore, one needs to express the differentials in ways which are invariant under coordinate transformations, or at least have definite and easily calculable transformation properties. It is useful to start by formulating the laws in fixed Cartesian coordinate systems, and then consider invariance under:- (i) Translations (ii) Rotations (iii) Boosts to Inertial Reference Frames (iv) Boosts to Accelerating Reference Frames Quantities such as scalars and vectors have definite transformation proper- ties under translations and rotations. Scalars are invariant under rotations. Vectors transform in the same way as a displacement under rotations. 5 2 Vector Analysis 2.1 Vectors Consider the displacement vector, in a Cartesian coordinate system it can be expressed as −! r =e ^x x +e ^y y +e ^z z (1) wheree ^x,e ^y ande ^z, are three orthogonal unit vectors, with fixed directions. The components of the displacement are (x; y; z). In a different coordinate system, one in which is (passively) rotated through an angle θ with respect to the original coordinate system, the displacement vec- tor is unchanged. However, the components (x0; y0; z0) defined with respect to 0 0 0 the new unit vectorse ^x,e ^y ande ^z, are different. A specific example is given by the rotation about the z-axis −! 0 0 0 0 0 0 r =e ^x x +e ^y y +e ^z z (2) The new components are given in terms of the old components by 0 x0 1 0 cos θ sin θ 0 1 0 x 1 @ y0 A = @ − sin θ cos θ 0 A @ y A (3) z0 0 0 1 z Hence, −! 0 0 0 0 r =e ^x ( x cos θ + y sin θ ) +e ^y ( y cos θ − x sin θ ) +e ^z z (4) The inverse transformation is given by the substitution θ ! − θ, −! 0 0 0 0 r =e ^x ( x cos θ − y sin θ ) +e ^y ( y cos θ + x sin θ ) +e ^z z (5) −! Any arbitrary vector A can be expressed as −! A =e ^x Ax +e ^y Ay +e ^z Az (6) wheree ^x,e ^y ande ^z, are three orthogonal unit vectors, with fixed directions. The components of the displacement are (Ax;Ay;Az). The arbitrary vector transforms under rotations exactly the same way as the displacement −! 0 0 0 0 A =e ^x ( Ax cos θ + Ay sin θ ) +e ^y ( Ay cos θ − Ax sin θ ) +e ^z Az (7) 6 2.2 Scalar Products Although vectors are transformed under rotations, there are quantities asso- ciated with the vectors that are invariant under rotations. These invariant quantities include:- (i) Lengths of vectors. (ii) Angles between vectors. These invariant properties can be formulated in terms of the invariance of a scalar product. The scalar product of two vectors is defined as −! −! A: B = Ax Bx + Ay By + Az Bz (8) The scalar product transforms exactly the same way as a scalar under rotations, and is thus a scalar or invariant quantity. −! −! A: B = Ax Bx + Ay By + Az Bz 0 0 0 0 0 0 = Ax Bx + Ay By + Az Bz (9) 2.3 The Gradient The gradient represents the rate of change of a scalar quantity φ(−!r ). The gra- dient is a vector quantity which shows the direction and the maximum rate of change of the scalar quantity. The gradient can be introduced through consid- eration of a Taylor expansion @φ @φ @φ φ(−!r + −!a ) = φ(−!r ) + a + a + a + ::: x @x y @y z @z −! = φ(−!r ) + −!a : r φ(−!r ) + ::: (10) The change in the scalar qunatity is written in the form of a scalar product of the vector displacement −!a given by −! a =e ^x ax +e ^y ay +e ^z az (11) with another quantity defined by −! @φ @φ @φ rφ =e ^ +e ^ +e ^ (12) x @x y @y z @z The latter quantity is a vector quantity, as follows from the scalar quantities φ(−!r ) and φ(−!r + −!a ) being invariant. Thus, the dot product in the Taylor ex- −! pansion must behave like a scalar. This is the case if rφ is a vector, since the scalar product of the two vectors is a scalar. 7 The gradient operator is defined as −! @ @ @ r =e ^ +e ^ +e ^ (13) x @x y @y z @z The gradient operator is an abstraction, and only makes good sense when the operator acts on a differentiable function. The gradient specifies the rate of change of a scalar field, and the direction of the gradient is in the direction of largest change. An example of the gradient that occurs in physical applications is the rela- tionship between electric field and the scalar potential −! −! E = − r φ (14) in electro-statics. This has the physical meaning that a particle will move (ac- celerate) from regions of high potential to low potential,.