Overview

Prerequisites

• MA1115
• MA2121

Learning Outcomes

Upon successful completion of this course, one should be able to:

In the area of Sequences and Infinite Series:

• State the difference between a sequence and an infinite series, and why convergence of a sequence of terms does not guarantee convergence of the series.
• Given an appropriate infinite series, determine whether it converges by the comparison test or integral test.
• For a series of functions, describe the difference between pointwise and uniform convergence, and how this is related to the continuity of the limit function.

In the area of Fourier Series:

• Determine whether a function, given either graphically or analytically, will have a Fourier series.
• Given an appropriate function, find its Fourier series expansion.
• Interpret the Fourier coefficients an and bn in terms of amplitude, phase, and power at specific frequencies.
• Convert between Fourier sine/cosine, real amplitude/ phase, and complex forms, and describe the utility of each representation.
• Use the Fourier series to find the response of a second order constant coefficient system to a periodic forcing function and interpret the relation between the coefficients in the response and those in the input.

In the area of Partial Differential Equations:

• Solve the wave equation in one or two-dimensional rectangular coordinates, with Dirichlet or Neumann boundary conditions, using Fourier series.
• Given a one-dimensional wave equation in an infinite medium, express the D'Alembert solution and display graphically how terms like f(x -ct) represent moving waves.
• Interpret the eigenvalues of the wave equation in terms of fundamental modes and frequencies of vibration, and, given a one or two-dimensional wave equation, determine its fundamental frequencies and modes.
• Explain the importance of characteristics in wave propagation, and, given a one-dimensional wave equation, find the equation of the characteristics.
• Sketch the general behavior of the ordinary Bessel functions of order 1, 2, 3.
• State the orthogonality integral for Bessel functions.
• Solve the wave equation in cylindrically symmetrical regions and determine the fun­damental frequencies and modes of propagation.

In the area of Fourier Transforms:

• State the definition of the complex Fourier transform, and the Fourier inversion formula.
• State conditions under which a function will have a Fourier transform, and given an appropriate function, compute its Fourier transform from the definition.
• Reduce the Fourier transform to either the sine or cosine transform in cases of appro­priate symmetry.
• Describe the relation between the Fourier transform and the Laplace transform, in terms of the solutions to second order constant coefficient ordinary differential equa­tions.
• Define the transfer function, and given a constant coefficient ordinary differential equation, determine its Fourier transform transfer function.
• State the convolution theorem for Fourier transforms and explain its importance in terms of the result from passing a signal through a sequence of linear systems.
• Graph and interpret, in physical terms, the amplitude and phase of a complex Fourier transform.
• Using appropriate tables to determine the inverse of a given Fourier transform, to include those cases where a change of variable must be performed to agree with formulas in the tables.
• State the transform formulas for the Fourier sine and Fourier cosine transforms.