
• Credits:  3.00 
 • Schedule:  Thursday evenings, 6:40–9:10pm 
 • Classroom:  Gowan 400 
 • Number:  01LEC (1852) 
 • Capacity:  25 
  
 • Instructor:  Dr. René Doursat 
 • Email:  doursatcua.edu 
 • Phone:  (202) 3195160 
 • Office:  for now: Pangborn 208C (later, probably next week: Pangborn 121) 
 • Hours:  Wednesday & Friday, 11:00am–12:30pm 
  
 • T. A.:  Amie Lin 
 • Email:  13lincardinalmail.cua.edu 
The objective of this course is to provide an introduction to the mathematical
methods that will be needed in subsequent graduatelevel courses in engineering.
Emphasis is placed on understanding the concepts for solving first and secondorder
differential equations (ODEs), as opposed to extricating an answer from math
packages such as Mathematica or MATLAB. The philosophy of understanding the
underlying concepts of the math is emphasized throughout the course and is
extended to the concept of integration in the complex plane and the origination
of the Fourier/Laplace transforms as well as their inverse transforms.
Additionally, various special functions are examined including: Gamma, Beta,
Bessel, and Legendre (and associated) polynomials. The process of separation
of variables in partial differential equations (PDEs) is covered in the three
major coordinate systems with application to the Laplace, heat, and wave equations.
 Firstorder ODEs
 Secondorder ODEs, Wronskian
 Power series solutions, Bessel functions, Legendre polynomials
 Convergence, regular and essential singularities
 Method of Frobenius, associated Legendre equation
 Integration techniques, complex Z and Wplanes,
multivalued functions and branch cuts
 CauchyRiemann equations, deformation of contours,
integration about a singularity
 Taylor and Laurent expansions of f(z), Jordan's lemma,
Gamma and Beta functions
 Fourier series solutions for periodic functions,
Fourier and inverse Fourier transforms
 Laplace and inverse Laplace transforms
 Classification of PDEs
 Separation of variables in Cartesian, cylindrical, and spherical
coordinates
 Solution of model equations: Laplace's, wave and heat equations
 Extended discussion of Bessel functions to include development
of second solutions, asymptotic representations, Hankel functions
(cylindrical and spherical)
 The use of laptops, tablets, cell phones, cameras,
recorders, or other devices is not allowed during any of the meetings
(lectures and exams). Bring paper and pens!
 Homework will be assigned at the end of every class
(i.e. weekly, except during recess) and due at the beginning
of the next class. It will consist of exercises and practice questions
intended to test and affirm your knowledge of the course content.
Collaboration on homework is accepted but must be acknowledged
(cite your study partners), and you must write your solutions
individually. Your paper must be written neatly, organized, and
stapled together. Late homework will be penalized (10% per day).
 Exams will include one midterm during the
semester (see Schedule for tentative date) and one final.
Makeup exams will not be given except in cases of
documented emergency.
 Important dates:
 Last day to add/drop courses → Sep 6
 Midterm Exam → Oct 10
 Last day to withdraw → Nov 8
 Final Exam → Dec 12, 5:457:45pm
Both grading policy and scale are subject to change.
Failure in either the practice component
(homework) or the lecture component (quizzes and exams)
will result in an F for the whole course.
• Grading Policy
Homework  40% 
Midterm Exam  25% 
Final Exam  35% 

⇨

• Grading Scale
100 → 90  A, A 
89 → 80  B+, B, B 
79 → 70  C 
69 → 0  F 

This is a tentative schedule, subject to
readjustment depending on the time we actually spend
in class covering the topics. Please check again regularly.
Course notes (if any), assignments and exam solutions are posted on the
CUA Blackboard,
under "Math Analysis.../Course Documents" and ".../Assignments".
Permanent reading assignment: it is
assumed that you are familiar with the contents of the course notes
of all past meetings.
Week 
Date 
Topics & Course Notes
(moved to the CUA Blackboard, under
"Math Analysis.../Course Documents") 
Reading Assignments 
Homework & Exams
(in Blackboard,
".../Assignments") 
1 
Thu, Aug 29 
0. Preliminaries:
the binomial theorem, derivatives of the $\ln$ function,
Euler's theorem, the imaginary $i\!$, operations in complex
space, conversion between Cartesian form $a+ib$ and polar
form $re^{i\theta}$. Definition of linear and nonlinear
differential equations.
1. Firstorder differential equations:
Special case of separable variables. Solving the general
linear case with an integrating factor $\alpha(x)$.

• read Ch. 7, pp. 329339 
• HW1
• HW1 key

2 
Thu, Sep 5 
1. Firstorder differential equations: (cont'd)
case of exact equations, inexact equations using an
integrating factor $\mu(x)$ to restore exactness,
"pseudolinear" types $y'=f(ax+by+c)$, Bernoulli equations,
"homogeneousdegree" functions $P$ and $Q$ and balanced
equations. Discussion of explicit/implicit equations versus
explicit/implicit solutions.

• read FirstOrder ODEs
• read Paul's Notes: Intro
• read Paul's Notes: 1stOrder
(first 5 sections)

• HW2
• HW2 key

3 
Thu, Sep 12 
2. Secondorder differential equations:
Homogeneous linear equations with constant coefficients and
characteristic polynomial. General homogeneous linear
equations with one known solution and reduction of order to
find another solution. Inhomogeneous linear equations: $y$
missing and reduction to firstorder linear. General
inhomogeneous case: solving homogeneous version first, then
finding a particular solution by undetermined coefficients.

• read SecondOrder ODEs
• read Ch. 7, pp. 342345
• read Ch. 7, pp. 375378
• read Paul's Notes: 2ndOrder
(except "Fundam.", "Wronskian",
"Variation" and "Mechanical")

• HW3
• HW3 key

4 
Thu, Sep 19 
1.2. Reviewing and completing DE methods:
Firstorder: restoring exactness with an integrating factor
$\mu(y)$ or $x^n y^m$. Secondorder: solving the linear
inhomogeneous case: finding a particular solution by
variation of parameters. Using the Wronskian to demonstrate
linear independence and determine an unknown solution from
a known one. General case with $x$ missing.
3. Power series solutions:
Discussion of convergence and the radius of convergence.
Power series solutions of firstorder DEs.

• read Ch. 7, pp. 358363
• read Paul's Notes: 2ndOrder
("Fundamental", "Wronskian",
"Variation" and "Mechanical")

• HW4
• HW4 key

5 
Thu, Sep 26 
The Wave/Helmholtz equation:
general solutions by separation of variables. Cartesian
coordinates: obtaining planar waves. Cylindrical coordinates:
obtaining Bessel's differential equations. Spherical
coordinates: obtaining Legendre's equations.
3. Power series solutions: (cont'd)
Secondorder DEs: Development of Bessel function of order 0,
$J_0(x)$. Higherorder Bessel functions the first type
$J_\nu(x)$. Finding Bessel functions of the second type
(Neumann functions) $N_\nu(x)$ by reduction of order.

• read Ch. 9, pp. 414426
• read Riffe's Notes: Wave Eq.
(Lectures 1822)
• read Paul's Notes: Series
(except "Euler Equations")

• HW5
• HW5 key

6 
Thu, Oct 3 
3. Power series solutions: (cont'd)
Convergence issues for the power series solution of Legendre
at $\pm&space;1$. Resolution of convergence using truncated
(polynomial) solutions for integer parameter values.
Nonessential vs. essential singularities. Frobenius method.
Associated Legendre equation. Solution about the weak
singular points. Complete (convergent) solutions using
associated Legendre polynomials.
4. Other integration techniques: integrating
$\exp(ax^2)$, $\exp(x^\alpha)$, $\exp(x)/x$, introduction of a
convergence factor (as in the Laplace transform).

• read Ch. 1, pp. 6574

• MT Review

7 
Thu, Oct 10 
Midterm Exam (2h)



8 
Thu, Oct 17 
5. Analytic functions in the complex plane:
Cartesian and polar form of complex numbers, DeMoivre's
formula, discussion of the $z$plane and the $w$plane.
Multivalued functions and the need for branch cuts to
maintain singlevaluedness. Definition of analytic functions.
The CauchyRiemann equations. Cauchy's integral theorem.
Deforming contours. Integration about a singular point.
Poles of higher orders. Numerous worked examples explaining
these concepts. The "unit circle" method.



9 
Thu, Oct 24 
Taylor series expansion of $f(z)$. Laurent expansion of $f(z)$. Jordan's lemma.
More worked examples. The Gamma function $\Gamma$: relationship to factor notation,
integrals solvable using $\Gamma$ functions. The Beta function $\operatorname{B}$
and solution of related integrals.



10 
Thu, Oct 31 
Fourier series solution of periodic functions. The Fourier series in
complex exponential form. Worked examples. Development of the Fourier transform
using the complex exponential form of the Fourier series. Definition of the Fourier
Transform and the inverse Fourier Transform. Use of the complex plane for finding the
inverse transform. 2D and 3D Fourier transforms. Properties of
Fourier transforms: shifting, differentiation, integration.



11 
Thu, Nov 7 
Introduction of the delta function $\delta$ and its use in determining the
Fourier transform of $\sin$ and $\cos$ functions. The Laplace transform and its need for
certain unbounded functions. Definition of the inverse Laplace transform. Finding the
inverse Laplace transform by using the Bromwich integral. Application examples.



12 
Thu, Nov 14 
Diffusion/Heat and Laplace equations:
Separation of variables in Cartesian coordinates. Separation
of variables in cylindrical coordinates. Separation of
variables in spherical coordinates. Power series solutions.



13 
Thu, Nov 21 
More discussion of Bessel functions: negative orders, fractional
orders, linear independence of fractional orders. Definition of the second homogenous
solution when integer orders are used. Asymptotic representations of Bessel functions.
The need for Hankel functions (cylindrical coordinates) and spherical Hankel functions
(spherical coordinates). Worked examples, particularly in cylindrical coordinates.



14 
Thu, Nov 28 
Thanksgiving recess  no class 
15 
Thu, Dec 5 
Review



16 
Thu, Dec 12 5:457:45pm 
Final Exam (2h)



 Contributions of course to meeting the professional components:
This graduate level engineering mathematics class develops the core
competency of mathematics needed for a graduatelevel study of
electrical, mechanical, civil, chemical, computer, or biomedical
engineering in subsequent applied courses. This class contains a
thorough review of solutions for differential equations. Integration
into the complex plane is introduced and developed with
connections made to inverse Fourier and Laplace transforms.
Fourier and Laplace transforms are developed from first principles
so that the underpinnings of the theory are clear and students have
a deeper understanding of the concepts and are not simply solving
equations using tables of transforms. Separation of variables
in the three major coordinate systems is developed from first
concepts and applied to practical engineering equations such as
the Laplace, wave, and diffusion equations.
 Relationship of the course to program objectives:
This course attempts to meet the stated departmental objectives:
 An appreciation of the mathematical tools needed for graduate engineering studies
 A working knowledge of advanced mathematical techniques
 Expected learning outcomes:
Upon completion of the course, students should demonstrate the ability to:
 CO1: Solve first and secondorder differential equations
 CO2: Solve differential equations using power series
 CO3: Work with integration techniques using the complex plane
 CO4: Work with the Gamma and Beta functions
 CO5: Develop Fourier and Laplace transforms and inverse transforms
 CO6: Work with separation of variables applied to solving differential equations
 Course outcome/ABET outcome matrix:
The Matrix below shows how this course contributes covers the 11 ABET Outcomes:
 ABET 01  ABET 02  ABET 03 
ABET 04  ABET 05  ABET 06  ABET 07 
ABET 08  ABET 09  ABET 10  ABET 11 
CO1  X   X 
X  X  X  
   X 
CO2  X   X 
X  X  X  
   X 
CO3  X   X 
X  X  X  
   X 
CO4  X   X 
X  X  X  
   X 
CO5  X   X 
X  X  X  
   X 
CO6  X   X 
X  X  X  
   X 
 Outcome assessment:
The course employs the following mechanisms to assess the above learning outcomes:
 Homework is assigned and graded weekly to assess the level
of student understanding of topics covered during the week.
The learning outcomes are also exhibited through the results
of the several exams given during the semester and the final
examination.
 The instructor frequently asks students if they understand
the lectures.
 The overall assessment of the course is done through the
university's Student Course Evaluation process.
 Process of improvement:
The instructor continuously tries to improve the course as described as follows:
 The instructor frequently evaluates the student performance
on homework and exams, and reviews the suggestions made by
students during the semester. Then the instructor takes proper
steps (such as different approaches to difficult material) to
correct problems.
 The instructor is available after class for
additional discussion.
 At the end of each semester, the instructor meets with the
chairman to discuss improvement plans for the course based on
the university's Student Course Evaluation process.
