PHYSICS 201 (FALL 2009)
MIDTERM EXAM 1
Given: October 23, 4:00 pm
Due: October 24, 12:00 noon
Dear students:
To let you have the benefit of all possible aids at your disposal, this exam is a take-home
one. Butplease note that you CANNOT have ANY consultation with ANY other person --------- NONE AT ALL!
1. The real part u(x,y) of an analytic functionj(z) is given to be
/11'
sin 2x / ( cos2x + cosh 2y).
Knowing this, determine the imaginary part of this function.
What, do you think,.,is the form of the functionj(z)
in terms ofz?
2. Locate the singularities of the function cot (n z) / (z - a)2 , where a is a real number,
and evaluate the residues of this function at those singularities.
/".
3. Usipg the calculus of residues,. evaluate the integral
"
S
21l'
cos {ne- sinlJ}. exp (coslJ) de
o
for all integral values of n.
4. Show that, for -1 < P < 1 and -Jt < A < Jt, the integral
00
1
X
1+ 2 x.
f
tAx
c.er:r
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71
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si",
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f
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(Euler)
e" (. .A ).
..
..
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integral
SihX
x(
X
1-
+
6.. 2. )
dx
)
a being a real number.
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=
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