Complex Analysis - Maths KU

3.1 Differentiability and Analyticity 143
Then, finally, (1) gives
The limit is f ′ (z) =2z − 5.
f ′ 2z∆z +(∆z)
(z) = lim
∆z→0
2 − 5∆z
∆z
∆z(2z +∆z − 5)
= lim
∆z→0 ∆z
= lim (2z +∆z − 5).
∆z→0
Rules of Differentiation The familiar rules of differentiation in the
calculus of real variables carry over to the calculus of complex variables. If
f and g are differentiable at a point z, and c is a complex constant, then (1)
can be used to show:
Differentiation Rules
Constant Rules: d
d
c = 0 and
dz dz cf(z) =cf ′ (z) (2)
Sum Rule:
Product Rule:
Quotient Rule:
Chain Rule:
d
dz [f(z) ± g(z)] = f ′ (z) ± g ′ (z) (3)
d
dz [f(z)g(z)] = f(z)g′ (z)+f ′ (z)g(z) (4)
� �
d f(z)
=
dz g(z)
g(z)f ′ (z) − f(z)g ′ (z)
[g(z)] 2
(5)
d
dz f(g(z)) = f ′ (g(z)) g ′ (z). (6)
The power rule for differentiation of powers of z is also valid:
d
dz zn = nz n−1 , nan integer. (7)
Combining (7) with (6) gives the power rule for functions:
d
dz [g(z)]n = n[g(z)] n−1 g ′ (z), nan integer. (8)