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Fall 21-22 - Quiz 2 - MAT3003Complex Variables and PDE
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2. *
(1 Point)
𝐿𝑒𝑑 𝑓 (𝑧) = (𝑧 βˆ’ 2𝑖)2 (𝑧 + 3)3 𝑒𝑧
β„Žπ‘Žπ‘  2𝑖 𝑖𝑠 π‘§π‘’π‘Ÿπ‘œ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 2 π‘Žπ‘›π‘‘ βˆ’ 3 𝑖𝑠 π‘§π‘’π‘Ÿπ‘œ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 3
β„Žπ‘Žπ‘  2𝑖 𝑖𝑠 π‘§π‘’π‘Ÿπ‘œ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 3 π‘Žπ‘›π‘‘ βˆ’ 3 𝑖𝑠 π‘§π‘’π‘Ÿπ‘œ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 2
β„Žπ‘Žπ‘  βˆ’ 2𝑖 𝑖𝑠 π‘§π‘’π‘Ÿπ‘œ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 2 π‘Žπ‘›π‘‘ 3 𝑖𝑠 π‘§π‘’π‘Ÿπ‘œ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 3
β„Žπ‘Žπ‘  βˆ’ 2𝑖 𝑖𝑠 π‘§π‘’π‘Ÿπ‘œ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 3 π‘Žπ‘›π‘‘ 3 𝑖𝑠 π‘§π‘’π‘Ÿπ‘œ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 2
10/22/2021
3. Question *
(1 Point)
𝐿𝑒𝑑 𝑓 (𝑧) = (π‘§βˆ’sin𝑧)
𝑧3
β„Žπ‘Žπ‘  π‘Ÿπ‘’π‘šπ‘œπ‘£π‘Žπ‘π‘™π‘’ sinπ‘”π‘’π‘™π‘Žπ‘Ÿπ‘–π‘‘π‘¦ π‘Žπ‘‘ 𝑧 = 0
β„Žπ‘Žπ‘  π‘π‘œπ‘™π‘’ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 3 π‘Žπ‘‘ 𝑧 = 0
β„Žπ‘Žπ‘  π‘’π‘ π‘ π‘’π‘›π‘‘π‘–π‘Žπ‘™ sinπ‘”π‘’π‘™π‘Žπ‘Ÿπ‘–π‘‘π‘¦ π‘Žπ‘‘ 𝑧 = 0
β„Žπ‘Žπ‘  π‘π‘œπ‘™π‘’ π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿ 2 π‘Žπ‘‘ 𝑧 = 0
4. *
(1 Point)
𝐿𝑒𝑑 𝑓 (𝑧) = 𝑧2𝑧+1βˆ’2𝑧
β„Žπ‘Žπ‘  π‘Ÿπ‘’π‘ π‘–π‘‘π‘’π‘’ 23 π‘Žπ‘‘ 𝑧 = 2
β„Žπ‘Žπ‘  π‘Ÿπ‘’π‘ π‘–π‘‘π‘’π‘’ βˆ’ 21 π‘Žπ‘‘ 𝑧 = 2
β„Žπ‘Žπ‘  π‘Ÿπ‘’π‘ π‘–π‘‘π‘’π‘’ βˆ’ 23 π‘Žπ‘‘ 𝑧 = 0
β„Žπ‘Žπ‘  π‘Ÿπ‘’π‘ π‘–π‘‘π‘’π‘’ 21 π‘Žπ‘‘ 𝑧 = 0
5. Question *
(1 Point)
1+𝑒𝑧
𝐿𝑒𝑑 𝑓 (𝑧) = 𝑧cos𝑧+sin𝑧
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 0]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 0]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 0]
𝑅𝑒𝑠[𝑓 (𝑧),𝑧 = 0]
10/22/2021
=1
=0
= βˆ’1
=2
6. Question *
(1 Point)
𝑧𝑒𝑧
𝐿𝑒𝑑 𝑓 (𝑧) = (π‘§βˆ’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)
∫𝐢
𝑑𝑧
π‘§βˆ’3
π‘€β„Žπ‘’π‘Ÿπ‘’ 𝐢 : |𝑧 βˆ’ 2| = 5
2πœ‹π‘–
0
βˆ’2πœ‹π‘–
1
8. Question *
(1.5 Points)
1
𝑧2 +5
2πœ‹π‘– ∫𝐢 π‘§βˆ’3 𝑑𝑧 π‘€β„Žπ‘’π‘Ÿπ‘’ 𝐢 𝑖𝑠
14
3
0
12
10/22/2021
|𝑧| = 4
9. Question *
(1 Point)
π‘‡β„Žπ‘’ π‘£π‘Žπ‘™π‘’π‘’ π‘œπ‘“ π‘‘β„Žπ‘’ π‘–π‘›π‘‘π‘’π‘”π‘Ÿπ‘Žπ‘™ ∫1+𝑖
0 𝑦 βˆ’ π‘₯ βˆ’ 𝑖3π‘₯ 2 𝑑𝑧 𝑖𝑠
1βˆ’π‘–
1+𝑖
1 βˆ’ 3𝑖
1 + 3𝑖
10. Question *
(1.5 Points)
∫𝐢
𝑧𝑑𝑧
(9βˆ’π‘§2 )(𝑧+𝑖)
πœ‹
5
π‘€β„Žπ‘’π‘Ÿπ‘’ 𝐢 : |𝑧| = 2
2πœ‹π‘–
βˆ’ 10𝑖
πœ‹π‘–
5
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10/22/2021
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