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The proof of the converse — that any function whose real and imaginary components
satisfy the Cauchy–Riemann equations is diFerentiable — will be omitted, but can be
found in any basic text on complex analysis, e.g., [
3
,
65
,
118
].
Remark
: It is worth pointing out that equation (7.19) tells us that
f
satis±es
∂f/∂x
=
−
i
∂f/∂y
, which, reassuringly, agrees with the ±rst equation in (7.5).
Example 7.4.
Consider the elementary function
z
3
= (
x
3
−
3
xy
2
) + i (3
x
2
y
−
y
3
)
.
Its real part
u
=
x
3
−
3
xy
2
and imaginary part
v
= 3
x
2
y
−
y
3
satisfy the Cauchy–Riemann
equations (7.18), since
∂u
∂x
= 3
x
2
−
3
y
2
=
∂v
∂y
,
∂u
∂y
=
−
6
xy
=
−
∂v
∂x
.
Theorem 7.3 implies that
f
(
z
) =
z
3
is complex diFerentiable. Not surprisingly, its deriva-
tive turns out to be
f
′
(
z
) =
∂u
∂x
+ i
∂v
∂x
=
∂v
∂y
−
i
∂u
∂y
= (3
x
2
−
3
y
2
) + i (6
xy
) = 3
z
2
.
²ortunately, the complex derivative obeys all of the usual rules that you learned in
real-variable calculus. ²or example,
d
dz
z
n
=
n z
n
−
1
,
d
dz
e
cz
=
c e
cz
,
d
dz
log
z
=
1
z
,
(7
.
20)
and so on. The power
n
can be non-integral — or even, in view of the identity
z
n
=
e
n
log
z
,
complex, while
c
is any complex constant. The exponential formulae (7.14) for the complex
trigonometric functions implies that they also satisfy the standard rules
d
dz
cos
z
=
−
sin
z,
d
dz
sin
z
= cos
z.
(7
.
21)
The formulae for diFerentiating sums, products, ratios, inverses, and compositions of com-
plex functions are all identical to their real counterparts, with similar proofs. Thus, thank-
fully, you don’t need to learn any new rules for performing complex diFerentiation!
Remark

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