Analytic functions of a complex variable:Complex numbers. Equations to curves in the plane in terms of z and z* – The Riemann sphere and stereographic projection. Analytic functions of z and the Cauchy-Riemann conditions. The real and imaginary parts of an analytic function – The derivative of an analytic function. Power series as analytic functions. Convergence of power series;Calculus of residues:Cauchy’s integral theorem. Singularities—removable singularity, simple pole, multiple pole, essential singularity. Laurent series. Singularity at infinity. Accumulation point of poles. Meromorphic functions. Cauchy’s integral formula – Solution of difference equations using generating functions and contour integration – Summation of series using contour integration. Evaluation of definite integrals using contour integration – Contour integration. Mittag-Leffler expansions of meromorphic functions – Causal, linear, retarded response. Dynamic susceptibility. Symmetry properties of the dynamic susceptibility. Dispersion relations. Hilbert transform pairs – Subtracted dispersion relations. Admittance of an LCR circuit. Discrete and continuous relaxation spectra – Analytic continuation and the gamma function:Definition of the gamma function and its analytic continuation. Analytic properties – Connection with gaussian integrals. Mittag-Leffler expansion of the gamma function. Logarithmic derivative of the gamma function. Infinite product representation for the gamma function. The beta function. Reflection and duplication formulas for the gamma function;Mobius transformations:Conformal mapping. Definition of a Mobius transformation and its basic properties – Fixed points of a Mobius transformation. The cross-ratio. Normal form of a Mobius transformation. Iterates of a Mobius transformation – Classification of Mobius transformations. The isometric circle. Group properties; the Mobius group. The Mobius group over the reals. The modular group. The invariance group of the unit circle. Connection with the pseudo-unitary group SU(1,1). The group of cross-ratios.
Multivalued functions; integral representations:Branch points and branch cuts. Algebraic and logarithmic branch points, winding point. Riemann sheets – Contour integrals in the presence of branch points. An integral involving a class of rational functions. Contour integral representation for the gamma function – Contour integral representations for the beta function and the Riemann zeta function. Connection with Bernoulli numbers. Zeroes of the zeta function. Statement of the Riemann hypothesis – Contour integral representations of the Legendre functions of the first and second kinds. Singularities of functions defined by integrals. End-point and pinch singularities, examples. Singularities of the Legendre functions. Dispersion relations for the Legendre functions;Laplace transforms:Definition of the Laplace transform. The convolution theorem. Laplace transforms of derivatives. The inverse transform, Mellin’s formula. The LCR series circuit. Laplace transform of the Bessel and modified Bessel functions of the first kind. Laplace transforms and random processes: the Poisson process – Laplace transforms and random processes: biased random walk on a linear lattice and on a d-dimensional lattice;Fourier transforms:Fourier integrals. Parseval’s formula for Fourier transforms. Fourier transform of the delta function. Relative `spreads’ of a Fourier transform pair. The convolution theorem. Generalization of Parseval’s formula. Iterates of the Fourier transform operator – Unitarity of the Fourier transformation. Eigenvalues and eigenfunctions of the Fourier transform operator – The Fourier transform in d dimensions. The Poisson summation formula. Some illustrative examples. Generalization to higher dimensions;Fundamental Green function for the laplacian operator:Green functions. Poisson’s equation. The fundamental Green function for the laplacian operator. Solution of Poisson’s equation for a spherically symmetric source – The Coulomb potential in d dimensions. Ultraspherical coordinates. A divergence problem. Dimensional regularization. Direct derivation using Gauss’ Theorem – The Coulomb potential in d = 2 dimensions.
The diffusion equation:Fick’s laws of diffusion – Diffusion in one dimension: Continuum limit of a random walk. The fundamental solution. Moments of the distance travelled in a given time – The fundamental solution in $d$ dimensions. Solution for an arbitrary initial distribution. Finite boundaries. Solution by the method of images. Diffusion with drift. The Smoluchowski equation. Sedimentation – Reflecting and absorbing boundary conditions. Solution by separation of variables. Survival probability – Capture probability and first-passage-time distribution. Mean first passage time;Green function for the Helmholtz operator; nonrelativistic scattering:The Helmholtz operator. Physical application: scattering from a potential in nonrelativistic quantum mechanics. The scattering amplitude; differential – cross-section – Green function for the Helmholtz operator;Total cross-section for scattering. Outgoing wave Green function for the Helmholtz operator. Integral equation for scattering – Green function for the Helmholtz operator;Exact formula for the scattering amplitude. Scattering geometry and the momentum transfer. Born series and the Born approximation. The Yukawa and Coulomb potentials. The Rutherford scattering formula;The wave equation:Formal solution for the causal Green function of the wave operator. The solution in (1+1) and (2+1) dimensions – The Green function in (3+1) dimensions. Retarded solution of the wave equation. Remarks on propagation in spatial dimensions d > 3. Differences between even and odd spatial dimensionalities. Energy-momentum relation for a relativistic free particle. The Klein-Gordon equation and the associated Green function – Rotations of the coordinate axes. Orthogonality of rotation matrices. Proper and improper rotations. Generators of infinitesimal rotations in 3 dimensions. Lie algebra of generators. Rotation generators in 3 dimensions transform like a vector – The general rotation matrix in 3 dimensions – The finite rotation formula for a vector. The general form of the elements of U(2) and SU(2). Relation between the groups SO(3) and SU(2) – The 2-to-1 homomorphism between SU(2) and SO(3). The parameter spaces of SU(2) and SO(3). Double connectivity of SO(3). The universal covering group of a Lie group. The group SO(2) and its covering group. The groups SO(n) and Spin(n). Tensor and spinor representations. Parameter spaces of U(n) and SU(n) – A bit about the fundamental group (first homotopy group) of a space. Examples
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