1. UNIT-IV COMPLEX INTEGRATION
IMPORTANT QUESTION: PART-A
1. State Cauchy’s integral theorem (or) fundamental theorem?
2. State Cauchy’s integral theorem for derivatives?
3. Evaluate where C is the circle of unit radius and centre at z= 1.
4. Evaluate if C is |z| = 2.
5. Evaluate if C is |z| = 1.
6. Evaluate if C is |z-1| = 2.
7. State Taylor’s series up to n-terms ?
8. Expand at z = 1 in Taylor’s series.
9. Expand Laurent’s series?
10. Define Isolated singular point?
11. Define Essential singular point?
12. Define Removable singular point?
13. State the nature of the singularity of f(z) = .
14. Find the zeros of f(z) = .
15. The function f(z) = find the pole and its order.
16. State the nature of the function f(z) = .
17. Find the residue of the function f(z) = at a simple pole.
18. Obtain the residue of the function f(z) =
19. Find the residue of f(z) = at the singular point z = 1.
20. State Cauchy’s residue theorem?

2. UNIT-IV COMPLEX INTEGRATION
PRAT-B
1. Using Cauchy’s residue theorem, Evaluate where C is the circle |z| = 3.
2. Using Cauchy’s residue theorem, Evaluate where C is the circle |z| = 4
3. Using Cauchy’s residue theorem, Evaluate where C is the circle 1<|z| < 4.
4. Using Cauchy’s residue theorem, Evaluate where C is the circle
5. Using Cauchy’s residue theorem, Evaluate where C is the circle |z| = 3.
6. Evaluate
7. Evaluate , |a|<1 using contour integration.
8. Evaluate .
9. Evaluate , a<b<0.
10. Evaluate , a<b<0.
11. Evaluate .
12.P.T = ,a>b>0.
13.P.T
14. Evaluate .
15. Find the residue of f(z) = at z = ai.
16. Expand f(z) = log (1+z) as Taylor’s series about z = 0 if |z|<1.
17. Expand f(z) = cosz about z = in Taylor’s series.
18. Expand in Laurent series (i) 2 <|z| <3. (ii) |z|>3.
19. Expand f(z) = in Laurent series if (i) |z| <2 (ii) |z| >3 (iii) 2 < |z| < 3 (iv) 1 <|z+1|< 3.

3. 20. Find the Laurent series of f(z) = in 1 <|z| < 2.
21. Using Cauchy’s integral formula, Evaluate where C is the circle
22. Using Cauchy’s integral formula, Evaluate where C if |z+1-i|=2.
23. Using Cauchy’s integral formula, Evaluate where C if |z-2| = ½.
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