Assignment#2 (Solve the following questions with the help of Cauchy integral
theorem or Cauchy Integral Formula)
Questions
Answers
(i) ∫𝑒𝑧
𝑧2+𝜋2𝑑𝑧, 𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 |𝑧|
𝐶= 4
0
(ii) ∫sin 3𝑧
(𝑧+𝜋)
𝑐𝑑𝑧, 𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 |𝑧|= 5
0
(iii) ∫sin 𝜋𝑧2+cos 𝜋𝑧 2
(𝑧2−3𝑧+2) 𝑑𝑧, 𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 |𝑧|= 3
𝐶
4𝜋𝑖
(iv) ∫𝑒3𝑖𝑧
(𝑧+𝜋)3𝑑𝑧
𝑐, 𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 |𝑧 − 𝜋|= 3.2
0
(v) ∫𝑧3+𝑧+1
𝑧2−3𝑧+2 𝑑𝑧, 𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑙𝑙𝑖𝑝𝑠𝑒 4𝑥2+ 9𝑦2= 1
𝐶
0
(vi) ∫2𝑧2+𝑧
𝑧2−1 𝑑𝑧, 𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 |𝑧 − 1|= 1
𝐶
3𝜋𝑖
(vii) ∫𝑒𝑖𝑧
5
𝐶𝑑𝑧, 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠 1 + 𝑖, −1 + 𝑖, −1 − 𝑖
0
(viii) ∫𝑧−1
(𝑧+1)2(𝑧−2)𝑑𝑧, 𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑒𝑙𝑙𝑖𝑝𝑠𝑒 𝑥2
9+(𝑦−1)2
16 = 1
𝐶
−2𝜋𝑖
9
(ix) ∫𝑧2−2𝑧
(𝑧+1)2(𝑧2+4) 𝑑𝑧, 𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 |𝑧|= 1
𝐶
8𝜋𝑖
25
(x) ∫𝑧
(𝑧+3)2(𝑧+1)𝑑𝑧,
𝐶
𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒 𝑤𝑖𝑡ℎ 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠
1 − 𝑖, −3 + 𝑖, 1 + 𝑖, −3 − 𝑖
0
(xi) ∫2𝑧−1
𝑧3−𝑖𝑧2−𝑧2𝑑𝑧,
𝐶
𝑊ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑤𝑖𝑡ℎ 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑣𝑒𝑟𝑡𝑖𝑐𝑒𝑠.
−𝑖, 3 − 𝑖, 2𝑖, 3 + 2𝑖
0
(xii) ∫𝑒2𝑧+𝑧2
(𝑧−1)5
𝐶𝑑𝑧 , 𝑤ℎ𝑒𝑟𝑒 𝐶 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 |𝑧|= 2
4𝜋𝑒2𝑖
3
Questions
Answers
(xiii) ∫1−cos(2(𝑧−3))
(𝑍−3)3
𝐶𝑑𝑧 , 𝑊ℎ𝑒𝑟𝑒 𝐶: |𝑧 − 3|= 1
4𝜋𝑖
