Problems for 18.112 Mid 1 (Open Book)
Oct. 20, 2006
1. (10) Find all solutions z to equation z 3 = 8i.
2. (15) Evaluate the integral
1
z1= 2
dz
.
(1 z)3
3. (20) Evaluate the integral
ez + z
dz
z2
in the two cases: 1) = cfw_z : z = 1;
2) = cfw_z : z
Problem 1. [5+7+3 points]
Let (X , d) be a metric space and let f : X >
tandng>
t be continuous maps.
(a) Suppose f (x0) > 9(x0). Show that there exists 7 > 0 such that f > 9(y) for all
y E B7(I0).
R :.= aw709) > 0
{5,3 continuous , so w
Solution for 18.112 ps 4
1(Prob 3 on P130).
Method 1. We only need to prove that these functions has no limit as z tends to
innity. We can prove this by constructing two sequence cfw_zn and cfw_wn of complex
numbers such that
lim zn = lim wn =
n
n
but
18.100B/C: Fall 2010
Solutions to Practice Final Exam
1. Suppose for sake of contradiction that x > 0. Then 1 x > 0 because the product of two positive
2
quantities is positive. Thus x + 0 < x + x (because y < z implies x + y < x + z for all x), i.e.,
2
2
Problem 1. [5+5+5 points]
(aMmwmmme@mmmndummmmamnmamMMymwmqmm.
E CX i: compacf (F, aivcn any open cover Ecle/Aua
Ly 9P, gets 14ch (M'ih A day index yet),
One Can Find a. Finite Salaam E: U u wit11
AICA c Find: m Lid: A,
(b) Prove that finite sets are a
Chain Rule and Total Dierentials
1. Find the total dierential of w = x3 yz + xy + z + 3 at (1, 2, 3).
Answer: The total dierential at the point (x0 , y0 , z0 ) is
dw = wx (x0 , y0 , z0 ) dx + wy (x0 , y0 , z0 ) dy + wz (x0 , y0 , z0 ) dz.
In our case,
wx
Problem 1. [5+5+5 points]
Let (X , d) be a metric space.
(a) State the definition of a connected subset of X Via separated sets, as in Rudin.
ECX is cemented iF
E=Av3
. :5 = or 859
An6= 714 Q
An? 9'
(b) Let (X , d) be connected. Show that a subset A C X i
Solution for 18.112 ps 5
1(Prob 1(f) on P161).
Solution: The function
f (z) =
1
z m (1 z)n
has two poles, 0 is a pole of order m and 1 is a pole of order n. At these poles, we
have the following expansions via Taylor series
1
n(n + 1) 2
n(n + 1) (n + m 2)
Solution for 18.112 Mid 1
Problem 1.
Solution:
z 3 = 8ei/2 = z = 2ei( 6 +
2n
)
3
, 0 n 2,
= z = 2i or z = 3 i or z = 3 i.
Problem 2.
Method 1.
1
z1= 2
dz
=
(1 z)3
=
2
0
iei t/2
dt
(eit /2)3
i
8
2
2
ei2t dt
0
2it 2
= 4i
= 0.
e
2i
0
Method 2. Let f (z) 1.
Solution for 18.112 Mid 2
Problem 1.
Solution: The function
1
1
is analytic in C cfw_2ni, n Z, and has simple pole at points z = 2ni. Thus
there are three poles in the region bounded by , which correspond to n = 0, 1.
Moreover, at each pole z, the residue
GREEN'S THEOREN AND I T S APPLICATIONS
The d i s c u s s i o n i n 1 1 . 1 9

11.27 o f , Apostol i s n o t complete
n o r e n t i r e l y r i g o r o u s , a s t h e a u t h o r himself p o i n t s o u t .
We
give here a rigorous treatment.
nr e e n ' s
Problem 1. [15 points] Fix n E N and let f: [0, 1] >
1
Show that f is Riemann integrable on [0, 1], and that / f dx = 0.
0
t be defined by
L
2, ifx = for kodd :0 < k < 2,
0 otherwise.
. L_(¥,P=(x.)=Z ° ax) (xim) w
in are xéxi] 13
20 =7 L(F)=5FnLl
'UFl
Problems for 18.112 Mid 2 (Open Book)
Nov. 22, 2006
1. (15) Evaluate
ez
dz
1
where is the circle z = 9.
f
2. (30) Let f (z) be analytic in the whole plane and assume that Rez (z) 0 as
z . Prove: f is a constant.
(Hint: Use formula (66) valid for z < R
Least squares interpolation
1. Use the method of least squares to t a line to the three data points
(0, 0),
(1, 2),
(2, 1).
Answer: We are looking for the line y = ax + b that best models the data. The deviation
of a data point (xi , yi ) from the model i
Solution for 18.112 ps 6
1(Prob 1 on P193).
Solution: The partial product
n
1
1
k2
1
Pn =
1
k
k=2
n
=
k=2
n
=
k +1k 1
k
k
k=2
=
thus
1
k
1+
n+1
,
2n
1
n=2
1
n2
= lim Pn
n
n+1
n 2n
1
= .
2
= lim
2(Prob 2 on P193).
Method 1. Note that
n1
(1 z)Pn = (1 z)(1 +