Complex Analysis - Maths KU

6.2 Taylor Series 321
(ii) If you haven’t already noticed, the results in (6), (7), (12), (13),
and (14) are identical in form with their analogues in elementary
calculus.
EXERCISES 6.2 Answers to selected odd-numbered problems begin on page ANS-18.
In Problems 1–12, use known results to expand the given function in a Maclaurin
series. Give the radius of convergence R of each series.
1. f(z) = z
2. f(z) =
1+z
1
4 − 2z
1
3. f(z) =
(1 + 2z) 2
5. f(z) =e −2z
z
4. f(z) =
(1 − z) 3
6. f(z) =ze −z2
7. f(z) = sinh z 8. f(z) = cosh z
9. f(z) = cos z
2
10. f(z) = sin 3z
11. f(z) = sin z 2
12. f(z) = cos 2 z [Hint: Use a trigonometric
identity.]
In Problems 13 and 14, use the Maclaurin series for e z to expand the given function
in a Taylor series centered at the indicated point z0. [Hint: z = z – z0 + z0.]
13. f(z) =e z , z0 =3i 14. f(z) =(z − 1)e −3z , z0 =1
In Problems 15-22, expand the given function in a Taylor series centered at the
indicated point z0. Give the radius of convergence R of each series.
15. f(z) = 1
z
1
, z0 =1 16. f(z) = , z0 =1+i,
z
17. f(z) = 1
, z0 =2i
3 − z
1
18. f(z) = , z0 = −i,
1+z
z − 1
19. f(z) = , z0 =1
3 − z
1+z
20. f(z) = , z0 = i
1 − z
21. f(z) = cos z, z0 = π/4 22. f(z) = sin z, z0 = π/2
In Problems 23 and 24, use (7) find the first three nonzero terms of the Maclaurin
series of the given function.
23. f(z) = tan z 24. f(z) =e 1/(1+z)
In Problems 25 and 26, use partial fractions as an aid in obtaining the Maclaurin
series for the given function. Give the radius of convergence R of the series.
i
z − 7
25. f(z) =
26. f(z) =
(z − i)(z − 2i)
z2 − 2z − 3
In Problems 27 and 28, without actually expanding, determine the radius of convergence
R of the Taylor series of the given function centered at the indicated point.
27. f(z) = 4+5z
, z0 =2+5i 28. f(z) = cot z, z0 = πi
1+z2