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Math 241 Final Exam Fall 2007

1. The radius of convergence for the Taylor series of f(z) = 1/(z-2)2 about the point z0 = i is 3. 2. The coefficient of the z-1 term in the Laurent series expansion of f(z) = (z-2)3/(z-1)2 in the region 1 < |z - 1| is 3. 3. The constant k such that the function v(x,y) = x3 - 3xy + ky is a harmonic conjugate of the function u(x,y) = x - xy2 is 1.

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58 views4 pages

Math 241 Final Exam Fall 2007

1. The radius of convergence for the Taylor series of f(z) = 1/(z-2)2 about the point z0 = i is 3. 2. The coefficient of the z-1 term in the Laurent series expansion of f(z) = (z-2)3/(z-1)2 in the region 1 < |z - 1| is 3. 3. The constant k such that the function v(x,y) = x3 - 3xy + ky is a harmonic conjugate of the function u(x,y) = x - xy2 is 1.

Uploaded by

jadest6
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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1.

Find the radius of convergence for the Taylor series of ( )


8
1
1
f z
z
=


about the point 2 2 2 2 z i = + .
(A) 1 (C) 2 (E) 3 (G) 4

(B)
3
2
(D)
5
2
(F)
7
2
(H)




2. Consider the Laurent series for the function
( )
2
2 3
2
z z
f z
z
+
=


in the region 1 1. z > What is the coefficient of the ( )
2
1 z

term?
(A) 6 (C) 3 (E) 1 (G) 3
(B) 4 (D) 0 (F) 2 (H) 6



3. Find the constant k such that the function ( )
2 3
, 3 1 v x y x y ky x = + + is a harmonic
conjugate of the function ( )
3 2
, 3 . u x y x xy y = +
(A) 3 (C) 1 (E) 1 (G) 3
(B) 2 (D) 0 (F) 2 (H) 4


4. Evaluate
2
0
i
iz
e dz
}
.
(A)
( )
2
1 i e

(C)
2
1 ie

(E)
2
i e

(G)
2
1 e


(B)
( )
2
1 i e

+ (D)
2
1 ie

+ (F)
2
i e

+ (H)
2
1 e

+




Math 241 Final Exam
Fall 2007

5. Evaluate





(A)
10

(C)
6

(E)
3

(G) 1
(B)
8

(D)
4

(F)
2

(H)


6. Evaluate
2
2 cos
0
d


}
.
(A) (C)
2
3

(E)
3

(G)
4


(B)
2
3

(D)
2

(F)
3

(H)
6




7. Evaluate
2
6 13
dx
x x

}
.
(A) (C)
24

(E)
24

(G)
2


(B)
12

(D) 0 (F)
12

(H)











8. In the Fourier series expansion of ( )
2
2 1 f x x = on (-1,1) find the coefficient on the ( ) cos 4 x
term.
(A) 0 (C)
1
2
(E)
1
2
(G) 1
(B)
2
1
2
(D)
2
2

(F)
2

(H) 2

9. Consider the Sturm-Liouville problem defined on 0
2
x

:
( ) 0 0 0, 0
2
y y y y

| |
+ = = =
|
\
.
Find all eigenvalues , 0,1, 2,
n
n = .
(A)
2
n
n = (C)
4
n
n
= (E)
( ) 2 1
4
n
n


= (G)
( )
2
2 1
2
n
n


=
(B)
2
4
n
n
= (D)
( ) 2 1
2
n
n


= (F) ( )
2
2 1
n
n = (H)
( )
2
2 1
4
n
n


=


10. The solution ( ) , u x t defined for 0 2, 0 x t to the wave equation
( ) ( ) with boundary conditions 0, 2, 0 is
tt xx x x
u u u t u t = = =

( ) ( ) ( ) ( )
2 2 2
0
, cos sin cos
n n n
n n
n
u x t A t B t x

=
( = +

.
Find
1 1
,
3 2
u
| |
|
\
with initial conditions ( ) ( ) ( ) ( ) , 0 3cos and , 0 2cos 3 .
t
u x x u x x = =
(A)
1

(C)
3

(E)
2
3
(G)
1
2

(B)
2

(D)
1
3
(F)
1
3
(H) 2












11. Let ( ) , u x t be a function defined for 0 , 0 x t such that
2
t xx x
u u u = +

with boundary conditions ( ) ( ) 0, , 0 for all 0 u t u t t = =
and initial condition ( ) ( ) , 0 sin 2
x
u x e x

= .
Use separation of variables to find ( )
4
,1 u

.
(With separation constant , you will find non-trivial solutions when 1 > , say
2
1 = + )
(A)
2
1
e


(C)
2
5
e


(E)
4
1
e


(G)
4
5
e



(B)
2
2
e


(D)
2
10
e


(F)
4
2
e


(H)
4
10
e







12. Consider a circular plate of radius 1 whose circular edge is maintained at the temperature
( ) 1, . u = The steady-state temperature ( ) ( ) ( )
0
1
, cos sin
n
n n
n
u r A r A n B n

=
( = + +


satisfies
2
1 1
0.
rr r r
r
u u u

+ + = Find the coefficient of the ( ) sin 3 term.


(A) 0 (C)
2
3
(E)
3

(G)
2
3


(B)
1
3
(D)
2
3

(F)
3

(H)
2
3




SOLUTIONS:
1. E
2. G
3. C
4. A
5. H
6. B
7. G
8. B
9. F
10. E
11. G
12. D

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