Exercise : 4.2 (Taylor's and laurent's series)

EXERCISE : 4.2

TAYLOR'S AND LAURENT'S SERIES

1. Expand f (z) = e^2z / ( z – 1) about z = 1 using Taylor's series expansion.

2. Obtain the expansion of log (1 - z), when | z | < 1 using Taylor's series expansion.

 [Ans. f (z) = -z – z² / 2 – z³ / 3 - … ∞ ]

3. Find the Taylor's series of f (z) = tanh z about the point z = 0.

 [Ans. f (z) = z / 1 – z³ / 3 + … ]

4. Expand the following in Laurent's series :

5. If f (z) = z + 4 / (z + 3) (z - 1)² find the Laurent's series expansion in the region.

 (i) 0 < | z - 1| < 4 (ii) | z – 1 | > 4

6. Find the Laurent's series expansion of f (z) = z / (z² + 1 ) )z² – 1)2 ]

(i) 1 < z < 2 (ii) | z | > 2

7. Find the Laurent's expansion of f (z) = 2²e^1/z about z = 0.

[Ans. f (z) z² + 1/z + 1/2! + 1/z 1/3! + ∞]

8. Obtain the Laurent's Series for f (z) = 1 / (z + 2) (1 + z²) in

(i) | z | < 1 (ii) 1< | z | < 2 (iii) | z |  > 2

9. Find the Laurent's series expansion of the function

f(z) = 1/ z (1 - z)² and specify the regions in which those expansions are valid.

10. Obtain the Taylor's series for f (z) = 2z³ + 1 / z(z + 1) about z = i

11. Expand the following functions in Taylor's series

12. Expand f(z) = z² / (z + 2) (z − 3) in a Laurent's series expansion if

 (i) | z | < 2 and (ii) 2 < | z | < 3

13. Find first four terms of the Laurent's series expansion valid in the region 0 <| z - 1| < 1 for the function f(z) = 2z + 1 / z³ + z² – 2z

14. Expand  as a Laurent’s  series.

17. Find the Laurent's expansion for f (z) = 1 / z(1 – z)2 in the regions

18. Find the Laurent's expansion of

19. (a) Find the Laurent's expansion of 7z – 2 / z (z+1) (z+2) in

20. Find the Laurent's series for f (z) = 6z + 5 / z (z − 2) (z+1) in the  region 1 < | z + 1 | < 3.

[A.U A/M 2018 R-17]

21. Find the Laurent's series for (z - 1) sin 1/z about z = 0

22. Find the Laurcnt's series for 1 – cos z / z about z = 0

[Ans. z / 2! – z3 / 4! + z5 / 6! - … ]

23. Find the Laurent's series for 1/z₃ e^z2 about z = 0

24. Find the Laurent's series for z^-1 e^-2z about z = 0

25. Expand 1 / z² – 3z + 2 in the region

26. Obtain the expansion of the function f(z) = z – 1 / z² in Taylor's series in powers of (z – 1) ) and give the region of validity and Laurent's series for the domain |z - 1| > 1 [Anna, May 1997]

27. Expand f (z) = 1 /  z (1-z) as a Laurent series

28. Find the Laurent series expansion of ƒ (z) = 1 / z² + 4z + 3 valid in the regions | z | < 1 and 0 < z + 1| < 2 [A.U A/M 2017 R-13]