Math 3210
Due Nov 16
Assignment #8
1. Outline a proof of Gauss Mean Value Theorem.
2. Explain how Maximum Modulus Principle follows from Gauss Mean Value Theorem.
3. Evaluate the contour integral.
sin(z 2 ) + cos(z 2 )
(a)
dz
(z 1)(z 2)
|z|=3
Hint: Use pa
Math 3210
Answers
Assignment #9
1. Write down the most important statements you know about series representation for analitic functions. Type your favorite formula from the course in the Discussion (Message) Board at my web
page.
http:/www.math.mun.ca/ mk
Math 3210
Answers
Assignment #7
1. Write the statement of the Cauchy-Goursat Theorem. Read the proof of the Cauchy-Goursat
Theorem (p.144-149). Write down the main steps of the proof.
This question was not marked. The Theorem and the prove are in the book
Math 3210
Answers
Assignment #6
1. Give an example (with explanaition) of
(a) simple closed non-dierentiable curve;
A triange or a square: no self-intersections, closed, not dierentible at the corners.
(b) dierentiable but not smooth arc;
An arc given by
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x plane ,Jrkem R2) is consTanT WWW};wa 41m 91m.
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f vx (n21) has ML lens-rm m0. 75MB,
Math 3210
Answers
Assignment #5
1. Find all values and the principal value of the complex expression
(a) (2)2/ = e2/(log(2) = e2/(ln 2+i+2ni) , P.V. = 41/ e2i
(b) (1 + i)i1 = e(i1)(ln 2+i/4+2ni)
(c) (1 i 3)5/2 = e5/2(ln 2i/3+2ni) , P.V. = 25/2 e5i/6
(d) T
Math 3210
Due Thur Oct 14
Assignment #4
1. Show that function v(x, y) is harmonic and nd the function u(x, y) of which it is harmonic conjugate.
(a) v = 2y 3x2 y + y 3
Answer: vxx + vyy = 6y + 6y = 0 thus it is harmonic.
u(x, y) = 2x x3 + 3xy 2 + C. Check
1
FUNDAMENTALS
10
Factoring
As above, if we are given (x
2)(x + 2), we can multiply it out to get
(x
2)(x + 2) = x2
4
Conversely, sometimes we are given an expression like x2 4 and asked to nd out if there
are two smaller polynomials which multiply togeth
1
FUNDAMENTALS
If n is not written, n = 2 ie
7
p
2
x=
p
x. If n is even, we must have a
0 and b
0.
p
Example: 4 = 2 since 22 = 4. In general, if x > 0 and n is an even number, there are two
p
p
numbers that satisfy y n = x, those being y = n x and y = n x
1
Fundamentals
1.1
Real Numbers
To start, well review the numbers that make up the real number system.
1. Natural numbers (denoted N) - these are just positive whole numbers:
1, 2, 3, . . .
2. Integers (denoted Z) - all natural numbers together with their
1
FUNDAMENTALS
4
There is a mixed type of interval called a half-open (or half-closed!) interval. These
intervals include one endpoint and not the other. For example:
[a, b) = cfw_x | a x < b
a
b
x
h
x
h
3
the half-open interval [a, b)
The last case is wh
Math 3210
Solutions
Assignment #3
Denition 1 Complex Derivative.
f (z) = lim
z0
f (z + z) f (z)
,
z
z C.
Denition 2 Continuity of complex function at point z0 .
lim f (z) = f (z0 ).
zz0
1. Using denition, nd derivative of z n , for
(a) any positive intege
Math 3210
Exercise only
Assignment #10
1. Write an essay [ Interesting, Amazing, Weird, Funny] Facts Known About Analytic Functions. (about 2 pages) You may be allowed to use it during the Final Exam.
2. Compose a True/False or multi-choise question about
Math 3210
Due Tue Nov 10
Assignment #7
1. Write the statement of the Cauchy-Goursat Theorem. Outline the proof.
2. Write the statement of the Cauchy Integral Formula. Outline the proof.
3. Write the statement of the Morera theorem. Outline the proof.
4. E
Math 3210
Due Thur Oct 7
Assignment #3
Denition 1 Complex Derivative.
f (z + z) f (z)
,
z0
z
f (z) = lim
z C.
Denition 2 Continuity of complex function at point z0 .
lim f (z) = f (z0 ).
zz0
1. Using denition, nd derivative of z n , for
(a) any positive i
Math 3210
Due Thur Sept 30
Assignment #2
1. Find domain of the following functions and rewrite them in the form u(x, y) + iv(x, y)
1
zz 5
1
(b)
z+i
1
(c) z +
z
(d) z + z 3
(a)
2. Use de Moivre Identity to show that for any natural n
n
cosnk sink
cos(n)
Math 3210
Due Nov 30
Assignment #10
1. Write an essay [ Interesting, Amazing, Weird, Funny] Facts Known About Analytic Functions.
(about 2 pages)
This question will not be marked but you may be allowed to use it during the Final Exam.
2. Pretend that you
MATHEM ATICS 3210
INTRODUCTION TO COM PLEX ANALYSIS
MATH 3210: Introduction to Complex Analysis
There was a young and adventurous man who found among his great-grandfathers papers a piece of torn
parchment that revealed the precise location of a hidden tr
Math 3210
Answers
Assignment #10
1. Write an essay [ Interesting, Amazing, Weird, Funny] Facts Known About Analytic Functions.
(about 2 pages)
This question will not be marked but you may be allowed to use it during the Final Exam.
2. Pretend that you are
Math 3210
Due Nov 17
Assignment #8
1. Explain how the fundamental theorem of algebra follows from the Liouvilles theorem.
2. Explain how Maximum Modulus Principle follows from Gauss Mean Value theorem.
3. Evaluate the contour integral.
cosh(z)
dz.
|z|=3 (
Math 3210
Due Tue Nov 3
Assignment #6
1. Draw the curve in the complex plane and classify it.
You options are: simple (no self intersections); dierentiable (z (t) is continuous); smooth
(dierentiable and z (t) = 0); contour (union of nite number of smooth
Math 3210
Answers
Assignment #8
1. Outline a proof of Gauss Mean Value Theorem.
This question was not marked. The proof is in the book and lecture notes. Please, come to discuss
if you have questions.
2. Explain how Maximum Modulus Principle follows from
Math 3210
Due Nov 24
Assignment #9
1. Write down Laurent or Taylor series expansion for a function f (z). What statements do
you know about their convergence? What are the properties of the function to which they
converge?
2. Let
n
n=0 bn z0
= S and
n
n=0
1
FUNDAMENTALS
13
Simplifying Rational Expressions
Just like when dealing with regular fractions, you can cancel a common term from the
numerator and denominator as long as it is not zero:
AC
if C 6= 0
BC
Example:
x2 1
(x 1)(x + 1)
x+1
=
=
2+x
+ 2)
x
2 1