Symmetry transformations and invariances. Examples. Definitions of group, subgroup and invariant subgroup. Continuous and discrete groups. Homomorphisms and isomorphisms. Representation of groups. Reducible and irreducible representations. Generators of continuous groups and their properties.
Structure constants. The groups SO(2), SO(3) and SU(2). Lorentz and Poincaré groups.
2. Complex Analysis – Analytical properties:
Revision of basic concepts. Complex functions and functions of complex variable. Cauchy-Riemann conditions and derivation of complex functions. Cauchy’s integral theorem. Contour integration. Taylor and Laurent’s serial expansion of a complex function. Analytical continuation. Mapping of Z plane in the W plane. Examples.
3. Complexanalysis – Calculation of residues:
Poles,singularities and branch points. Residue Theorem. Calculation of the residue of a pole of order n. Cauchy principal value. Calculation of several defined integrals.
4. Differential equations:
Resolution of linear differential equations using the variable separation method, generic presentation.
First order linear differential equations; example of an RLC circuit. Resolution of Helmotz equation using the variable separation method, in Cartesian, cylindrical and spherical coordinates. Singularities in differential equations. Resolution of differential equations using the serial expansion method. Examples: harmonic oscillator, Bessel’s equation.
5. The Dirac δ function:
Presentation and properties. Representation of the Dirac δ function by sequences of functions. Problems concerning the Mathematical interpretation of Dirac δ function. Different representations of Dirac δ function.
6. Green functions:
Presentation of Green’s functions using Laplace and Poisson equations. Physical interpretation and properties. Usefulness of Green’s functions in different domains of physics.
7. Fourier Series:
Presentation of the concept, orthogonality and completeness relations. Expansions in Fourier series of sines and cosines. Examples. Convergence of Fourier series, integration and differentiation.
8. Integral transforms:
Fourier and Laplace transforms. Fourier transforms in series of sines and cosines. The Fourier integral.The inversion theorem. Application of Fourier transforms – resolution of finite impulse in sinusoidal waves. Fourier transforms of derivatives. Application to the resolution of differential equations.
9. Special functions:
Legendre functions. The generator function of Legendre polynomials, explicit form of these polynomials and examples of application. Recurrence relations. Properties. Rodriguez formula. Associated Legendre functions. Spherical harmonics. Properties and usefulness. Hermite polynomials. Generator function, recurrence relations, properties and application. Laguerre polynomials and Laguerre functions.
Bibliography of reference
ARFKEN, G.; and WEBER, H. J. (1995). Mathemathical Methods for Physicists. New York: Academic Press.
MATHEWS, J. and WALKER, R. L.; BENJAMIN, W. A. (1965). Mathematical Methods of Physics. Menlo Park, California.