Department of Mathematics
Christopher Newport University
Math 355-01 Complex Variables Spring Term 2009
Homework # 4
Due Friday Feb.21.09
1. Apply the definition (3), Sec.19, of Derivative to give a direct proof that
f0(z) = 1
(1 + z)2when f(z) = z
1 + z
2. Use an appropriate theorem to show that f0(z) does not exist at any point:
(a) f(z) = z+z2, z 6= 0 (b) f(z) = excosy−i exsiny.
3. Use an appropriate theorem to show that each of the functions is differentiable in the indicated domain
then find f0(z).
(a) f(z) = sinhxsiny−icoshxcosy, for all z
(b) f(z) = e−2θcos(2ln r) + i e−2θsin(2ln r),for r > 0,0< θ < π.
4. Show that each of these functions is entire and find f0(z).
(a) f(z) = (2 x+x2−y2) + i2y(x+ 1) (b) f(z) = −cosxsinhy+isinxcoshy