264 CHAPTER 7 • TAYLOR AND LAURENT SERIES
-------... EXERCISES FOR SECTION 7.2
l. By computing derivatives, find the Maclaurin series for each function and state
where it is valid.(a) sinh z.
(b) cosh z.
(c) Log(l+z).- Using methods other than computing derivatives, find the Maclaurin series for
(a) cos^3 z. Hint: Use the trigonometric identity 4cos^3 z = cos3z + 3cosz.
(b) Arctan z. Hint: Choose an appropriate contour and integrate second
series in Equations (7-12).
(c) f (z) = (z^2 + l)sinz.(d) f (z) = e= cosz. Hint: cosz = ~ (e" + e-"), so f (z) = ~e(l+i)z +
~e<^1 -^1 >=. Now use the Maclaurin series for e•.- Find the Taylor series centered at a = 1 and state where it converges for
(a) f (z) = !:::;.
(b) f (z) = :-3^1 -•. Hint: =-3^1 - • = (!) 2 1-• - l ,i = (l ) 2 (z -1) t-^1 9'.
- Let f (z) = •^1 ~· and set f (0) = 1.
(a) Explain why f is analytic at z = 0.
(b) Find the Maclaurin series for f ( z).
(c) Find the Maclaurin series for 9 (z) = f cf((,) d(,, where C is t he straight.
line segment from 0 to z.- Show that f (z) = 1 ~. has its Taylor series representation about the point °' = i
given by
(^00) (z - i)"
f(z) = ~ (1-ir+1 • for all z E D~(i) = {z: lz - i l < J2}.
6. Let f (z) = (1 + z)^13 = exp (.8 Log (1 + z)) be the principal branch of (1 + zl,
where .B is a fixed complex number. Establish the validity for z E D 1 (0) of the
binomial expansion(^1 + z)fJ = l + .Bz + .B (.8 - l ) z^2 + .B (.8 - l) (.8 -(^2) ) z3 +..
2! 3!
_ ~ ,B(,8- 1 )(.8-2)··· (,8-n+ 1) n
-l+L..J n. I z.
n=l