FORMULA SHEET FOR COMPLEX VARIABLES FINAL EXAM
Fact 1: Standard Taylor expansions of analytic functions are: [They may be used without
explanation, except to say where they are claimed to hold; or except if one is asked to derive
one directly from the the
Polytechnic Institute of NYU
MA 3112
Final
December 13, 2010
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Signature:
ID #:
Instructor/Section:
Directions: You have 120 minutes to answer the following questions. You must show all
your work as neatly and clearly as possible and indicate th
Calculus 1
WKST- Related Rates
Name:_
Date:_
Find the rate of change of the distance between the origin and a moving point on the graph of y x 2 1
if dx dt 2 cm/sec.
1.
x 2 x 2 1 2
x
r 2
x 2 x 2 1 2
4
3 x 2 1
12
r
1 4
x 3x 2 1 1 2 4 x 3 6 x dx dr
2
dt
BIRDS CENSUS*
County Environmental Committee
4
15
44
8
31
13
68
5
30
9
W
ee
k
20
61
41
21
75
81
Monthly Total
5
3
W
ee
k
17
8
7
18
9
14
*Data collected by committee member volunteers.
8
W
ee
k2
10
12
4
15
10
69
Total Sighted
71
Red-headed Woodpeckers
2
Gr
Some theorems in complex variables: Part II - B
Recall that if an analytic function f (z) is analytic inside a circle of radius R about the center z = a,
that is, on the open disk
cfw_z | |z a| < R
then it is innitely dierentiable in this disk and moreove
Some theorems in complex variables: Part I - B
Recall that the Cauchy-Riemann equations read for f (z) = u(x, y) + iv(x, y), u = Re(f (z), v =
Im(f (z)
u
v
u
u
=
and
=
=
v
v
v
v
v
with derivative f (z) = u + i x . An alternate form of these equations is:
Polytechnic Institute of NYU
MA 3112
Midterm
November 15, 2010
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Directions: You have 120 minutes to answer the following questions. You must show all
your work as neatly and clearly as possible and indicate
MA 3112
Polytechnic University
MIDTERM
March 30 , 2007
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ID #:
Instructor/Section:
Directions: You have 90 minutes to answer the following questions. You must show
all your work as neatly and clearly as possible and indicate the nal a
_ , _g' \v [I
[aw/(ta HWR DavIM7 C
.wr _Vvv
1. Show that 2 +7. [E0 +1.).
A (a) exp(2 :l:3rri) = e2; (17} exP< 4 "V 2
(c) exp(: + xi) = exp :.
7 ' ' t that
5 Write lexp(2-: + [)1 and lexp(i:2)1 1n terms of .x and y. Then show
. 1' -2.rv
E)
SOME ANSWERS TO PRACTICE FINAL FOR COMPLEX VARIABLES:
# 1. Give an explicit example of each below:
1. An analytic function f (z ) with equals its Taylor expansion about z = 0 on the
open disk |z | < 3, but has a pole of order 3 at z = 3.
1
An answer : tr
Polytechnic Institute of NYU
MA 3112
Final
December 13, 2010
Print Name:
Signature:
ID #:
Instructor/Section:
Directions: You have 120 minutes to answer the following questions. You must show all
your work as neatly and clearly as possible and indicate th