Contents:
Analytic functions of a complex variable (Part I) : 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.
Analytic functions of a complex variable (Part II) : The derivative of an analytic function. Power series as analytic functions. Convergence of power series.
Calculus of residues (Part I) : 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.
Calculus of residues (Part II) : Solution of difference equations using generating functions and contour integration.
Calculus of residues (Part III) : Summation of series using contour integration. Evaluation of definite integrals using contour integration.
Calculus of residues (Part IV) : Contour integration. Mittag-Leffler expansions of meromorphic functions.
Linear response; dispersion relations (Part I) : Causal, linear, retarded response. Dynamic susceptibility. Symmetry properties of the dynamic susceptibility. Dispersion relations. Hilbert transform pairs.
Linear response; dispersion relations (Part II) : Subtracted dispersion relations. Admittance of an LCR circuit. Discrete and continuous relaxation spectra.
Analytic continuation and the gamma function (Part I) : Definition of the gamma function and its analytic continuation. Analytic properties.
Analytic continuation and the gamma function (Part II) : 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 (Part I) : Conformal mapping. Definition of a Mobius transformation and its basic properties.
Mobius transformations (Part II) : Fixed points of a Mobius transformation. The cross-ratio. Normal form of a Mobius transformation. Iterates of a Mobius transformation.
Mobius transformations (Part III) : 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 (Part I) : Branch points and branch cuts. Algebraic and logarithmic branch points, winding point. Riemann sheets.
Multivalued functions; integral representations (Part II) : Contour integrals in the presence of branch points. An integral involving a class of rational functions. Contour integral representation for the gamma function.
Multivalued functions; integral representations (Part III) : 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.
Multivalued functions; integral representations (Part IV) :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 (Part I) : 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 (Part II) : Laplace transforms and random processes: biased random walk on a linear lattice and on a d-dimensional lattice.
Fourier transforms (Part I) : 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.
Fourier transforms (Part II) : Unitarity of the Fourier transformation. Eigenvalues and eigenfunctions of the Fourier transform operator.
Fourier transforms (Part III) : The Fourier transform in d dimensions. The Poisson summation formula. Some illustrative examples. Generalization to higher dimensions.
Fundamental Green function for the laplacian operator (Part I) : Green functions. Poisson's equation. The fundamental Green function for the laplacian operator. Solution of Poisson's equation for a spherically symmetric source.
Fundamental Green function for the laplacian operator (Part II) : 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 (Part I) : 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 diffusion equation (Part II) : 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.
The diffusion equation (Part III) : Reflecting and absorbing boundary conditions. Solution by separation of variables. Survival probability.
The diffusion equation (Part IV) : Capture probability and first-passage-time distribution. Mean first passage time.
Green function for the Helmholtz operator; nonrelativistic,scattering (Part I) : 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;
nonrelativistic scattering (Part II) : Total cross-section for scattering. Outgoing wave Green function for the Helmholtz operator. Integral equation for scattering.
Green function for the Helmholtz operator;
nonrelativistic scattering (Part III) : 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 (Part I) : Formal solution for the causal Green function of the wave operator. The solution in (1+1) and (2+1) dimensions.
The wave equation (Part II) : 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.
The rotation group and all that (Part I) : 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 rotation group and all that (Part II) : 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 rotation group and all that (Part III) : 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.
Lecture 1: Mod-06 Lec-14 Multivalued functions; integral representations (Part I)
Lecture 2: Mod-10 Lec-26 The diffusion equation (Part II)
Lecture 3: Mod-01 Lec-02 Analytic functions of a complex variable (Part II)
Lecture 4: Mod-10 Lec-25 The diffusion equation (Part I)
Lecture 5: Mod-05 Lec-13 Möbius transformations (Part III)
Lecture 6: Mod-01 Lec-01 Analytic functions of a complex variable (Part I)
Lecture 7: Mod-13 Lec-36 The rotation group and all that (Part III)
Lecture 8: Mod-09 Lec-24 Fundamental Green function for Δ2 (Part II)
Lecture 9: Mod-05 Lec-12 Möbius transformations (Part II)
Lecture 10: Mod-13 Lec-35 The rotation group and all that (Part II)
Lecture 11: Mod-09 Lec-23 Fundamental Green function for Δ2(Part I)
Lecture 12: Mod-05 Lec-11 Möbius transformations (Part I)
Lecture 13: Mod-13 Lec-34 The rotation group and all that (Part I)
Lecture 14: Mod-08 Lec-22 Fourier transforms (Part III)
Lecture 15: Mod-04 Lec-10 Analytic continuation and the gamma function (Part II)
Lecture 16: Mod-08 Lec-21 Fourier transforms (Part II)
Lecture 17: Mod-12 Lec-33 The wave equation (Part II)
Lecture 18: Mod-08 Lec-20 Fourier transforms (Part I)
Lecture 19: Mod-04 Lec-09 Analytic continuation and the gamma function (Part I)
Lecture 20: Mod-12 Lec-32 The wave equation (Part I)
Lecture 21: Mod-07 Lec-19 Laplace transforms (Part II)
Lecture 22: Mod-03 Lec-08 Linear response; dispersion relations (Part II)
Lecture 23: Mod-07 Lec-18 Laplace transforms (Part I)
Lecture 24: Mod-11 Lec-31 Green function for (Δ2 + k2); nonrelativistic scattering (Part III)
Lecture 25: Mod-03 Lec-07 Linear response; dispersion relations (Part I)
Lecture 26: Mod-11 Lec-30 Green function for (Δ2 + k2); nonrelativistic scattering (Part II)
Lecture 27: Mod-06 Lec-17 Multivalued functions; integral representations (Part IV)
Lecture 28: Mod-02 Lec-06 Calculus of residues (Part IV)
Lecture 29: Mod-11 Lec-29 Green function for (Δ2 + k2); nonrelativistic scattering (Part I)
Lecture 30: Mod-02 Lec-05 Calculus of residues (Part III)
Lecture 31: Mod-06 Lec-16 Multivalued functions; integral representations (Part III)
Lecture 32: Mod-10 Lec-28 The diffusion equation (Part IV)
Lecture 33: Mod-02 Lec-04 Calculus of residues (Part II)
Lecture 34: Mod-06 Lec-15 Multivalued functions; integral representations (Part II)
Lecture 35: Mod-10 Lec-27 The diffusion equation (Part III)
Lecture 36: Mod-02 Lec-03 Calculus of residues (Part I)