# Dr. Abdel Rahman Yousef ## Taught Courses All Items        Course    Description Academic Year    Classification; some  physical  models  (heat, wave, Laplace  equations); separation of variables; Sturm-liouville BVP; Fourier series and Fourier transform; BVP involving rectangular and circular regions; special functions (Bessel and Legendre); BVP involving cylindrical and spherical regions.
Third Year

Real numbers: order, absolute value, bounded subsets, completeness property, Archimedean property; supremum and infimum; sequences: limit, Cauchy sequence, recurrence sequence, increasing, decreasing sequence,  lim sup,  lim inf  of a sequence; functions: limit,  right, left limit, continuity at a point, continuity on an interval; uniform continuity (on an interval) relations between continuity and uniform continuity, differentiability: definition, right, left derivative, relation between differentiability and continuity, Rolle’s theorem, mean value theorem, applications on mean value theorem.
Second Year

Evolution of some mathematical concepts, facts and algorithms in arithmetic, algebra, trigonometry, Euclidean geometry, analytic geometry and calculus through early civilizations, Egyptians, Babylonians, Greeks, Indians, Chinese, Muslims and Europeans, evolution of solutions of some conjectures and open problems.
Fourth Year

Complex numbers: geometric interpretation, polar form, exponential form: powers and roots; regions in the  complex  plane;  analytic  functions; functions of complex variables: exponential and logarithmic functions ; trigonometric and hyperbolic functions;  definite integrals; Cauchy theorem; Cauchy integral formula; Series; convergence of sequence and series, Taylor series; Laurrent series; uniform convergence; integration and differentiation of power series, zeros of analytic functions; singularity ; principle part; residues; poles; residue theorem of a function; residues at poles; evaluation of improper integrals; integration through a branch cut.
Fourth Year