Unit iv complex integration

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Unit iv complex integration

  1. 1. UNIT-IV COMPLEX INTEGRATIONIMPORTANT QUESTION: PART-A1. 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. 2. UNIT-IV COMPLEX INTEGRATION PRAT-B1. 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| = 43. 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 circle5. Using Cauchy’s residue theorem, Evaluate where C is the circle |z| = 3.6. Evaluate7. 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.T14. 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. 3. 20. Find the Laurent series of f(z) = in 1 <|z| < 2.21. Using Cauchy’s integral formula, Evaluate where C is the circle22. 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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