274 CHAPTER 7 • TAYLOR AND LAURENT SERIES
• EXAMPLE 7.8 Find the Laurent series representation for f (z) =^00 ',.!-^1
that involves powers of z.
Solution We use the Maclaurin series for cosz - 1 to write
1 2 14 16
f (z) = - 2fz + ;rrzz4 - filz + ....
We formally divide each term by z^4 to obtain the Laurent series
- 1 1 z^2
f (z) = 2z2 + 24 - 720 + · ·. (valid for z "# O).
- EXAMPLE 7.9 Find the Laurent series for exp(-~) centered at a = O.
00 nSolution The Maclaurin series for exp z is exp z = L ~ 1 , which is valid for all
n = O
00 / - !)"z. We let -z-^2 take the role of z in this equation to get exp ( ~) = n~O #,
whicli is valid for lzl > 0.-------~EXERCISES FOR SECTION 7.3- Find two Laurent series expansions for f (z) = , 3 2_, 4 that involve powers of z.
- Show that f (z) = 1 2_, = 1 ~. i -Ef bas a Laurent series representation about the
point ZQ = i given by
1 00 (1 -i)"-^1
f (z) = - = - ~. n
1 - z n = l L., (z -i) (valid for lz -ii > v'2).- Find the Laurent series for f (z) = s;~.2· that involves powers of z.
00
4. Show that !:; = - I: (•-'IJ" is valid for lz - 11 > l. Hint: Refer to the solution
n-=-0
for Exercise 3(a), Section 7.2.- Find the Laurent series for sin U) centered at a = 0. Where is the series valid?
00
6. Show that !:; = - I: (•_:~)" is valid for lz -l l > 2. Hint: Use the hint for
n=O
Exercise 3(b), Section 7.2.- Find the Laurent series for f (z) = cosh :-cosz that involves powers of z.
- Find the Laurent series for f (z) = •'< 1 '._,) 2 that involves powers of z and is valid
for lzl > 1. Hint: (i-'n• = (t~:l,.