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MAT 342 Applied Complex Analysis Final Examination
May 15, 2007 Hand back this text with your bluebook, your name on each. Show all your work in your bluebook! Total score = 200 You may quote theorems we have learned this semester, but for ful
Some limit problems: solutions. The method is always the same. The problem is of the form
za
lim f (z) = L.
Unraveling the denition of limit, this means that given any > 0 we can produce a > 0 so that |f (z) L| < whenever |z a| < . We proceed as follows:
Midterm 1 Solutions
Note that there dierent forms of this test; yours may be slightly dierent from this one. 1. (a) (15 points) What are the 4 fourth roots of 9? Write 9 as 9ei . Then the rule for n-th roots gives the four roots as i( + 2k ) 4 9e 4 4 , k
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Problem 1 2 3 Total Score
MAT 342 Applied Complex Variables Midterm 1
February 27, 2007
CALCULATOR AND CELLPHONE POLICY: No calculators or computers may be used on this text. NO CELLPHONES are permitted in the examination room. Show all your w
MAT 342 Applied Complex Analysis Midterm 2
April 12, 2007 SOLUTIONS 1. (a) (12 points) Using the denition ez = ex cos y + iex sin y, where z = x + iy, show that the function f (z) = ez is analytic. SOLUTION: It is enough, writing ez as u(x, y) + iv(x, y),
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Problem 1 2 3 4 Total Score
MAT 342 Applied Complex Analysis Midterm 2
April 12, 2007
Show all your work on these pages! Total score = 100 You may quote theorems we have learned this semester, but for full credit you must state them carefully,
MAT 342 Applied Complex Analysis
Spring 2016 Midterm Exam Example
Solutions
1.
a) 1)
i arctan
2 3 1 + i(2 + i3 ) = 2 3 1 + 2i + i4 = 2 3 + 2i = 12 + 4e
i arctan 1
= 4e
3
2
2
= 4ei 6
2)
2
10
+ cos
+ i sin
+ sin
1 + cos
3
3
2
2
4
4
= 1 + cos
+ i sin