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Fall 21-22 - Quiz 2 - MAT3003Complex Variables and PDE
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(1 Point)
𝐿𝑒𝑡 𝑓 (𝑧) = (𝑧 − 2𝑖)2 (𝑧 + 3)3 𝑒𝑧
ℎ𝑎𝑠 2𝑖 𝑖𝑠 𝑧𝑒𝑟𝑜 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 2 𝑎𝑛𝑑 − 3 𝑖𝑠 𝑧𝑒𝑟𝑜 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 3
ℎ𝑎𝑠 2𝑖 𝑖𝑠 𝑧𝑒𝑟𝑜 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 3 𝑎𝑛𝑑 − 3 𝑖𝑠 𝑧𝑒𝑟𝑜 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 2
ℎ𝑎𝑠 − 2𝑖 𝑖𝑠 𝑧𝑒𝑟𝑜 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 2 𝑎𝑛𝑑 3 𝑖𝑠 𝑧𝑒𝑟𝑜 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 3
ℎ𝑎𝑠 − 2𝑖 𝑖𝑠 𝑧𝑒𝑟𝑜 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 3 𝑎𝑛𝑑 3 𝑖𝑠 𝑧𝑒𝑟𝑜 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 2
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3. Question *
(1 Point)
𝐿𝑒𝑡 𝑓 (𝑧) = (𝑧−sin𝑧)
𝑧3
ℎ𝑎𝑠 𝑟𝑒𝑚𝑜𝑣𝑎𝑏𝑙𝑒 sin𝑔𝑢𝑙𝑎𝑟𝑖𝑡𝑦 𝑎𝑡 𝑧 = 0
ℎ𝑎𝑠 𝑝𝑜𝑙𝑒 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 3 𝑎𝑡 𝑧 = 0
ℎ𝑎𝑠 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 sin𝑔𝑢𝑙𝑎𝑟𝑖𝑡𝑦 𝑎𝑡 𝑧 = 0
ℎ𝑎𝑠 𝑝𝑜𝑙𝑒 𝑜𝑓 𝑜𝑟𝑑𝑒𝑟 2 𝑎𝑡 𝑧 = 0
4. *
(1 Point)
𝐿𝑒𝑡 𝑓 (𝑧) = 𝑧2𝑧+1−2𝑧
ℎ𝑎𝑠 𝑟𝑒𝑠𝑖𝑑𝑢𝑒 23 𝑎𝑡 𝑧 = 2
ℎ𝑎𝑠 𝑟𝑒𝑠𝑖𝑑𝑢𝑒 − 21 𝑎𝑡 𝑧 = 2
ℎ𝑎𝑠 𝑟𝑒𝑠𝑖𝑑𝑢𝑒 − 23 𝑎𝑡 𝑧 = 0
ℎ𝑎𝑠 𝑟𝑒𝑠𝑖𝑑𝑢𝑒 21 𝑎𝑡 𝑧 = 0
5. Question *
(1 Point)
1+𝑒𝑧
𝐿𝑒𝑡 𝑓 (𝑧) = 𝑧cos𝑧+sin𝑧
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 0]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 0]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 0]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 0]
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=1
=0
= −1
=2
6. Question *
(1 Point)
𝑧𝑒𝑧
𝐿𝑒𝑡 𝑓 (𝑧) = (𝑧−1)
3
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 1]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 1]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 1]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 1]
= 3𝑒2
= 𝑒2
= 7𝑒2
= 5𝑒2
7. Question *
(1 Point)
∫𝐶
𝑑𝑧
𝑧−3
𝑤ℎ𝑒𝑟𝑒 𝐶 : |𝑧 − 2| = 5
2𝜋𝑖
0
−2𝜋𝑖
1
8. Question *
(1.5 Points)
1
𝑧2 +5
2𝜋𝑖 ∫𝐶 𝑧−3 𝑑𝑧 𝑤ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠
14
3
0
12
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|𝑧| = 4
9. Question *
(1 Point)
𝑇ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 ∫1+𝑖
0 𝑦 − 𝑥 − 𝑖3𝑥 2 𝑑𝑧 𝑖𝑠
1−𝑖
1+𝑖
1 − 3𝑖
1 + 3𝑖
10. Question *
(1.5 Points)
∫𝐶
𝑧𝑑𝑧
(9−𝑧2 )(𝑧+𝑖)
𝜋
5
𝑤ℎ𝑒𝑟𝑒 𝐶 : |𝑧| = 2
2𝜋𝑖
− 10𝑖
𝜋𝑖
5
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10/22/2021