Study Guide Analysis 2018

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Montclair State University
Department of Mathematical Sciences
Comprehensive Exam Study Guide – Analysis
The following list of topics is not meant to be comprehensive. The exam may
cover topics that are not listed on this study guide but were covered in the
corresponding course or covered in prerequisite undergraduate courses. The
sample questions are merely representative of content that may be covered
on the exam.

Real Variable
Topics
Real Number System: Ordered sets, fields, real and complex fields, Euclidean spaces, least upper-bound property, construction of R from Q.
Basic Topology: countable and uncountable sets, metric spaces, open and
closed sets, compact sets, perfect sets, connected sets.
Sequences and Series: Convergent sequences, Cauchy sequences, completeness, series, convergence criteria for series, absolute convergence, addition
and multiplication of series.
Continuity: Limits of functions, continuous functions, continuity and compactness, continuity and connectedness, discontinuities, monotonic functions, infinite limits and limits at infinity.
Differentiation: The derivative of a function, Mean-Value Theorem, continuity of derivatives, L’Hopital’s rule, derivatives of higher order, Taylor’s
Theorem.
Riemann-Stieljes Integral: Definition and existence of the Riemann-Stieljes
integral, properties of the Riemann-Stieljes integral, Fundamental theorem
of Calculus

References
Apostol, Tom M., Mathematical Analysis, Addison-Wesley
Bartle, R. G ., The Elements of Real Analysis, Wiley
DePree and Swartz, Introduction to Real Analysis, W iley
Fitzpatrick, Advanced Calculus, Pws Pub.
Parzynski and Zipse, Introduction to Analysis
Royden, H. L., Real Analysis, Prentice Hall
Rudin, Walter, Principles of Mathematical Analysis, McGraw Hill
Shilov, George E., Elementary Real and Complex Analysis, Dover

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Complex Variable
Topics
Complex Number System: algebra of complex numbers, geometry of the
complex plane, Euler’s formula, polar representation of complex numbers,
roots of unity
Elementary Functions of a Complex Variable: complex polynomials, exponential and trigonometric functions in the complex plane, logarithm in the
complex plane, branch points and branch cuts, elementary Riemann surfaces
Derivative: limits in the complex plane, definition of derivative in the complex plane, Cauchy-Riemann Equations
Integration: parameterization of curves, integration on a parameterized
curve, Cauchy’s Theorem, Cauchy’s Integral Formula, Maximum-Modulus
Theorem, Morera’s Theorem, Liouville’s Theorem
Series: complex power series, Taylor-Series expansion of an analytic function, Laurent series, isolated singularities of analytic functions
Calculus of residues: Residue Theorem, evaluation of integrals by Calculus of Residues, evaluation of real integrals as contour integrals in the complex plane, Argument Principle, Rouche’s Theorem

References
Ablowitz, M. J. and Fokas A. S. Complex Variables and Applications 2nd
Ed., Cabridge University Press (2003)
Silverman, R. Complex Analysis with Applications, Dover (2010)
Ahlfors, L. Complex Analysis 3d Ed., McGraw-Hill (1979)

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Sample Questions
1. Prove or provide a counter-example to the following statment:
If the functions {fn }∞
n=1 are continuous on [a, b] and
fn (x) → 0

n→∞

as

for all x ∈ [ab]. Then,
Z

b

fn (x) dx → 0 .
a

2. Give an example of a function that is continuous and bounded, but
not uniformly continuous on its domain.
3. Prove or provide a counter-example to the following statment:
If f : X → Y is continuous and D ∈ Y is compact, then f −1 (D) is
compact.
4. Prove or provide a counter-example to the following statment:
The set of all continuous functions on [0, 1] with
Z

1

|f − g|

ρ(f, g) =
0

is a metric space.
5. Let
f (z) = 2(x3 + iy 3 ) + (y 2 + x2 )
where z = x + iy.
Determine f 0 (z) for points z where the derivative is well-defined.
Determine the set z for which f (z) is analytic.
6. Compute the integral
eβz
dz
1 + z2

I
C

where β > 0 and C is the circle of radius
counter-clockwise orientation.
7. Evaluate

3
2

centered on z = 0 with a

Z
|z| dz
C

where C is
(a) the straight line from −i to i
(b) semicircular arc on |z| = 1 from −i to i with Re z ≥ 0
and explain the significance of your results.

3

8. Prove or provide a counter-example to the following statment:
If f (z) is an entire function such that Im f (z) = 0, then f (z) is constant.
9. f (z) =

z2
z−1

(a) Find a series expansion for f (z) centered on z = 0 that is valid
at z = 2 (e.g., converges to f (2) = 4).
(b) Describe the set on the complex plane where your series converges.
10. Evaluate the integral
Z
0

π

1
dθ.
sin θ + 1
2

4
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