Complex Variables II
PROBLEM SET 1
Due January 30, 2007
1.* Determine the pre-images of all straight lines through the origin in
the w -plane, under the mapping
w=i
zi
.
z+i
(Hint: Try rst the axes u = 0 and v = 0, then look at the general line v = u
for
Polar representation of complex numbers
modulus and argument of a complex number
We already know that r = sqrt(a2 + b2) is the modulus of a + bi. We write this modulus
as |a+bi|.
We also know that the point p(a,b) in the Gauss-plane is a representation of
Complex Variables II
PROBLEM SET 7
Due March 20, 2007
1. (a) If u(r, ) is a harmonic function for r < R, verify from the results of section 117 that u
satises the mean-value property
u(0) =
2
1
2
u(r, )d, r < R.
0
(b) Show that u also satises
1
r2
u(0) =
Complex Variables II
PROBLEM SET 5
Due February 27, 2007
1. Problem 2 page 396 of text. The R is as dened in problem 1 on this page, and as
discussed in class.
2. Problem 1, page 403 of text.
3.Problem 3, page 403. (Set x1 = 1, x2 = a etc.)
4.Problem 4, p
helley.pdf Complex Variables II
PROBLEM SET 8
Due April 3, 2007
1. Complete all the steps of the proof of lemma 3, chapter 1, of the lecture notes for
lectures 9-14.
2. Verify for step (iii) of the proof of the mapping theorem in the lecture notes:
G ( )
=
e
a
d
Joukowski irfoil, elta=0, psilon - .1,a =1
'1.5
0.5
-0.5
-1
- t.3
-z
-2.5
rh-rl-
.
-2
O1 .?*nr.
v
L
.
4
Y'
-1.5
-1
-0.5
0.5
2.5
t
global
eps
* c o s ( t h ) ) . / ( e p s ^ 2 + 2 * e p s .* ( 1 - e p s ) . * c o s ( t h ) + ( 1 - e p s ) ^ 2 )
x=e
Complex Variables II
PROBLEM SET 2
Due February 6, 2007
1-5. Problems 4, 6, 7,8, 10 pages 370-372 of text.
6.* (a) What is the image of the interior R of the half-disk
|z | < 1, Im(z ) > 0,
under the mapping Z = 1 z + 1/z ?
2
Indicate the images of the po
Complex Variables II
PROBLEM SET 3
Due February 13, 2007
1.(a) The ux of uid through an arc C connecting two streamlines = a and = b
is dened as
q nds.
Q=
C
Here n = (nx , ny ) is the unit normal to C , as shown in gure (1) below. Then (ny , nx is
the uni
Complex Variables II
PROBLEM SET 9
Due April 10, 2007
1. Consider the ODE zw + (2r + 1)w + zw = 0. Consider solutions represented by a contour integral
w = C ez v()d. Show that if C is as shown in gure 1 (beginning an ending at + along the Re()-axis),
and
Complex Variables II
PROBLEM SET 10 (Final problem set)
1. Verify from the integral that (1/2) = ( 1 )! =
2
following from (z + 1) = z (z ), show that
Due April 24, 2007
. Using only the analytic continuation of
1
3
3
5
1
3 1
4
,
,
.
!=
!=
! = 2 ,
!=
2
Complex Variables II
PROBLEM SET 4
Due February 20, 2007
1. Problem 7, page 387 of text.(Take A a positive constant.)
2. Problem 9, page 387 of text.(Take A a positive constant.)
3. The (counter-clockwise) moment acting on a cylinder in 2D, due to the pre
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