Summer 2022

1. The Riemann integral: Riemann sums, the fundamental theorem of calculus and the indefinite integral. Methods for computing integrals: integration by parts, substitution, partial fractions. Improper integrals and application to series. 2. Uniform and pointwise convergence. Cauchy criterion and the Weierstrass M-test. Power series. Taylor series. 3. First order ODE’s: initial value problem, local uniqueness and existence theorem. Explicit solutions: linear, separable and homogeneous equations, Bernoulli equations. 4. Systems of ODE’s. Uniqueness and existence (without proof). Homogeneous systems of linear ODE’s with constant coefficients. 5. Higher order ODE’s: uniqueness and existence theorem (without proof), basic theory. The method of undetermined coefficients for inhomogeneous second order linear equations with constant coefficients. The harmonic oscillator and/or RLC circuits. If time permits: variation of parameters, Wronskian theory.

1. Fields: the definition of a field, complex numbers.

2. Linear equations: elementary operations, row reduction, homogeneous and non-homogeneous equations, parametrization of solutions.

3. Vector spaces: examplex, subspaces, linear independence, bases, dimension.

4. Matrix algebra: matrix addition and multiplication, elementary operations, the inverse matrix, the determinant and Cramer’s law. Linear transformations: examples, kernel and image, matrix representation.

1. Complex valued-functions and the complex exponential. Fourier coefficients of piecewise continuous periodic functions. Basic operations and their effects on Fourier coefficients: translation, modulation, convolutions, derivatives.
2. Uniform convergence: Cesaro means, the Dirichlet and Fejer kernels, Fejer’s theorem. The Weierstrass approximation theorem for trigonometric polynomials and for polynomials. Uniqueness of Fourier coefficients. The Riemann-Lebesgue lemma. Hausdorff’s moment problem. Convergence of partial sums and Fourier series for $C^2$-functions.
3. Pointwise convergence: Dini’s criterion. Convergence at jump discontinuities and Gibbs phenomenon.
4. $L^2$-theory: orthonormal sequences and bases. Best approximations, Bessel’s inequality, Parseval’s identity and convergence in $L^2$.
5. Applications to partial differential equations: the heat and wave equations on an interval with constant boundary conditions, the Dirichlet problem for the Laplace equation on the disk, the Poisson kernel.

Bibliography:

• Korner, Fourier analysis
• Stein and Shakarchi, Fourier analysis