Download Midterm Topics for Midterm I

Midterm Topics for Midterm I
Math 128, Fall 2013
Instructor: Dr. Doreen De Leon
Midterm I will be given on Wednesday, September 25, 2013, during the scheduled class
time in the classroom, Science II, Room 207.
The exam will be closed-note and closed-book, and no calculators will be permitted.
Make sure to bring your student ID with you to the exam. Paper will be provided.
Topics:
I.
II.
III.
Complex Numbers and Their Properties – Sections 1-10 of text, class notes
A.
Modulus and properties of the modulus
B.
Argument of a complex number (and principal argument)
C.
Complex conjugates.
D.
Representations of complex numbers
1.
Cartesian.
2.
Polar.
3.
Exponential.
E.
Operations with complex numbers
1.
Addition.
2.
Multiplication.
3.
Inverse and division.
F.
Finding roots of complex numbers
Regions in the Complex Plane – Section 11 of text, handout and class notes
A.
Neighborhoods and deleted neighborhoods.
B.
Interior and exterior points.
C.
Boundary points.
D.
Open and closed sets; closure of a set.
E.
Connected sets.
F.
Domains.
G.
Bounded sets.
Functions of a Complex Variable – Sections 12-14, class notes
A.
Functions of a complex variable
1.
Representation in the form f(z) = u(x, y) + i v(x, y).
2.
Representation in the form f(z) = u(r, θ) + i v(r, θ).
B.
Mappings
1.
Quadratic function.
2.
Exponential function.
C.
Limits
1.
Properties.
2.
Determining the limit of a complex function.
3.
Limits involving infinity.
D.
Continuous functions – definition and properties
IV.
Analytic Functions – Section 15-25, class notes
A.
Derivative of complex functions
1.
Definition.
2.
Properties (including the chain rule).
3.
Determining the derivative of a given function using these
properties.
B.
Cauchy-Riemann equations
1.
Cauchy-Riemann theorem (the theorem given in class).
2.
Cauchy-Riemann equations in polar form.
3.
Using the Cauchy-Riemann theorem to
a)
Investigate where a given function is differentiable, and
b)
determine the derivative of a function in the region in
which it is differentiable.
C.
Analyticity
1.
Definition
2.
Determining whether (and where) a function is analytic.