Page 104 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 9: Taylor and Laurent Series

9.6 Review Questions Notes

1. Show that for all z,

 1
z
e  e  (z 1) . j

j 0 j!

 n 
j  for tanh z ?
2. What is the radius of convergence of the Taylor series  c z j 

 j 0 

3. Show that
1  (z i) j

1 z   j 1

j 0 (1 i)


for |z – i| < 2.
1
4. If f(z)  , what is f (i)?
(10)
1 z

5. Suppose f is analytic at z = 0 and f(0) = f’(0) = f’’(0) = 0. Prove there is a function g analytic
at 0 such that f(z) = z g(z) in a neighborhood of 0.
3
6. Find the Taylor series for f(z) = sin z at z = 0.
0
7. Show that the function f defined by
 sinz 0
 z for z 

f(z)  
 1 for z  0
 
is analytic at z = 0, and find f’(0).
8. Find two Laurent series in powers of z for the function f defined by

1
f(z) 
z (1 z)
2

and specify the regions in which the series converge to f(z).
9. Find two Laurent series in powers of z for the function f defined by

1
f(z) 
z(1 z )
2

and specify the regions in which the series converge to f(z).
1
10. Find the Laurent series in powers of z – 1 for f(z) = in the region 1 < |z – 1| < .
z

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