The problems are from complex variable class, 500 level in undergraduate.

1. Find f ΄(z) when

(a) f (z) = 3z2 – 2z + 4

(b) f (z) = (1- 4z2)3

(c) f (z) = (z – 1) / (2z + 1) (z ≠ -1/2)

(d) f (z) = (1 + z2)4 / z2 (z ≠ 0)

2. Show that

(a) a polynomial

P (z) = a0 + a1z + a2z2 + ··· + anzn (an ≠ 0)

of degree n (n ≥ 1) is differentiable everywhere, with derivative

P ΄(z) = a1 + 2a2z + ··· + nanzn-1

(b) the coefficients in the polynomial P (z) in part (a) can be written

p

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0

4

6

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x

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