Math 3080 - Solutions of Assignment 9 - Fall 2010.
1. Locate each of the isolated singularities of the given function, and tell whether it
is a removable singularity, a pole, or an essential singularity. If the singularity is
removable, nd the analytic ex
Assignment 1 for MATH/ECON3900
Rumseys Solutions
1. [2 marks] Let S be a function of t, let , K, be positive constants, and let dX be a random normal with
mean zero and variance dt. Then the SDE
dS = (K S) dt + S dX
is known a mean-reverting process, mean
MATH 3080 Complex Variables
Assignment 3
1. Use the denition of the derivative to dierentiate
(a) f (z) = z 2 + 2z;
1
(b) f (z) = z .
2. Verify the Cauchy-Riemann equations for
(a) u(x, y) = x3 3xy 2 and v(x, y) = 3x2 y y 3 ;
y
x
(b) u(x, y) = x2 +y2 and
MATH 3080 Complex Variables
Assignment 2
1. Find
(a)
(b)
(c)
(d)
(e)
the absolute values and principal arguments of
2 5i;
2.728 3.010i;
3 2i
5i+4 ;
cos 250 + i sin 250 ;
3 cos 4 + i sin 4 .
3
3
2. Find (1
p
i 3)85 (in the form a + ib).
3. Find the followi
MATH 3080 Complex Variables
Assignment 1
1. Use the denition of complex numbers to verify (1)(7) of Theorem 1.1.
2. Use the appropriate denitions to verify (1)(10) of Theorem 1.2.
3. Find the real and imaginary parts of
(a)
p
(b) (1 + i 3)2 ;
i+2
;
i 2
(c
Math 3080 - Solutions of Assignment 1 - Fall 2010.
1. Present a complete list of 7th roots of w = ( 1+i )4 .
2
Solution: Let us rst compute w.
1+i
w = ( )4 = (cos( ) + i sin( )4 = cos( ) + i sin( ) = 1.
4
4
2
The principal argument of w = 1 is , and its 7
Math 3080 - Solutions of Assignment 5 - Fall 2010.
1. Use the rules of dierentiation to nd f (z ) when
(
)
1. f (z ) = exp (1 4z 2 )3 .
2. f (z ) = sin(z ).
3. f (z ) = cos(z ).
4. f (z ) = (sin(z 2 )4 .
Solution of Part 1: Using the chain rule, we have
(
Math 3080 - Solutions of Assignment 4 - Fall 2010.
1. Establish the following relations for the complex trigonometric functions:
1. sin(z ) = sin(z ).
2. sin(z ) = sin(z ).
3. cos(z ) = sin(z ).
2
Solution of part 1: Let z = x + iy . Then
sin(z ) =
=
=
ei
Math 3080 - Solutions of Assignment 3 - Fall 2010.
1. Let a1 , a2 , . . . be a series of complex numbers. Prove that if
limn an = 0.
converges then
converges
n=1 an
Solution: We present two solutions for this question.
First solution: Recall the following
Math 3080 - Solutions of Assignment 2 - Fall 2010.
1. Let D1 and D2 be two domains with nonempty intersection. Show that D1 D2 is a
domain as well. Is D1 D2 a domain? If yes, support your answer with a proof; and
if no, present a counter-example.
Solution
Math 3080 - Solutions of Assignment 6 - Fall 2010.
1. Find the radius of convergence for the given power series.
(k!)2
k
1.
k=0 (2k)! (z 3) .
(1)n n
2.
z.
n=0 7
Solution of Part 1: We use the ratio formula as below.
1
= lim
R k
(k+1)!)2
(2(k+1)!
(k!)2
(
Math 3080 - Solutions of Assignment 7 - Fall 2010.
1. Evaluate the given integrals. Reference any theorem that you use.
z
)
1. e 2+sin(z8 , = cfw_z C : |z 1| = 2.
z 2z
2. z sin(z 2 )dz , where is any piecewise smooth simple curve joining i to .
z
3. e 3s
Complex Variables
Solutions for Midterm 2 - Fall 2010.
1.
(i) State Cauchy-Goursat Theorem including all of its conditions.
(ii) Dene a simply connected domain, and state the extension of Cauchy-Goursat
Theorem for simply connected domains.
Solution: Refe
Complex Variables
Midterm 1 - Fall 2010.
Time: 4:30-6:00 pm.
1. State the rst form of Greens Theorem including all of its conditions.
Solution: Refer to the notes.
2. Let a1 , a2 , . . . be a series of complex numbers. Prove that if
limn an = 0.
n=1 an
co
Math 3080 - Solutions of Assignment 10 - Fall 2010.
Use the Residue Theorem to compute the following integrals:
2
1. x x45x+1 dx.
+1
sin(2)
2. 54 sin() d.
cos
3. 0 x4 +3xx+2 dx. (Hint: See Example 3 Page 157.)
2
ln(x)
4. 0 (1+x2 )2 dx.
x2 ln(
5. 0 (1
Math 3080 - Solutions of Assignment 8 - Fall 2010.
1. Evaluate the following integral
1
2i
sin(3z )
dz,
z 16
where is the circle |z | = 2 oriented counter-clockwise.
Solution: The function f (z ) = sin(3z ) is analytic on C. Since is a piecewise
smooth si
MATH 3080
Introduction to Complex Variables
Assignment Instructions
Some general suggestions and rules for all assignments:
Please write clearly and legibly. Write in full sentences, and not just
strings of equations.
Handwritten is ne, unless you are u