complex and variables

1. Assignment-2 (Unit-2 )
1. Evaluate ∫ 𝑧2
𝑑𝑧𝑐
where C is the arc from 𝐴(1,1) to 𝐵(2,4) along (i) 𝑦 = 𝑥2
(ii) 𝑦 = 3𝑥 − 2
2. Evaluate ∫
𝑠𝑖𝑛𝜋𝑧2+𝑐𝑜𝑠𝜋𝑧2
(𝑧−1)(𝑧−2)𝑐
𝑑𝑧, where C is |𝑧| = 3. using Cauchy’s integral formula.
3. By using Cauchy’s integral formula, evaluate ∫
𝑧+1
𝑧4−4𝑧3+4𝑧2𝑐
𝑑𝑧, where C is the circle
|𝑧 − 2 − 𝑖| = 2.
4. Using Cauchy’s integral formula, evaluate ∫
7𝑧−1
𝑧2−3𝑧−4
𝑑𝑧𝑐
, where C is the ellipse 𝑥2
+
4 𝑦2
= 4.
5. Using Cauchy’s integral formula, evaluate ∫
𝑒2𝑧
(𝑧−1)(𝑧−2)
𝑑𝑧𝑐
,
where C is the circle |𝑧| = 3.