A&EP 321
Mathematical Physics
Prelim #2
November 20, 2003
7:309:30 p.m.
CLOSED BOOK
NO CALCULATORS
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1. For this problem use SchwartzChristoffel mapping techniques. Consider the
30%
mapping function
z
=
z
(
w
) with
dz
dw
=
1
(
w

1)
2
/
3
(
w
+ 1)
2
/
3
and the mapping of the real
w
axis onto the
z
plane.
a. Show that the expression
z
=
Z
ω

1
dx
(
x

1)
2
/
3
(
x
+ 1)
2
/
3
will satisfy the above differential equation.
b. Using the expression in part (a) above, locate the mapping of
the point
w
=
w
1
=

1 onto the
z
plane. Call this point
z
1
.
c. Let
w
=
w
2
= +1 with
w
2
mapping to
z
2
.
z
2
will lie along the
mapping of the line segment from
w
1
to
w
2
along the real
w
axis.
How does this line segment map onto the
z
plane? Note that so
far the length of this mapped line segment and consequently the
exact location of
z
2
in the
z
plane is unknown.
d. Show that the distance form
z
1
to
z
2
is given by:
Z
1
0
2
dx
(1

x
2
)
2
/
3
.
This integral is approximately equal to 4.
e. Now map the line segment along the real
w
axis from
∞
to
w
1
and the one from
w
2
to +
∞
onto the
z
plane. What does your
result say about the mapping of
w
=
±∞
onto the
z
plane?
2. Find the coefficients of the Laurent series for
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 Line segment, exponential Fourier series

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