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
realvariable 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 nonintegral — 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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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Power Series, Taylor Series, Peter J. Olver

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