The CauchyRiemann relations

24.2 THE CAUCHY–RIEMANN RELATIONS
Show that the function f(z) = 1/(1 − z) is analytic everywhere except at z = 1.
Since f(z) is given explicitly as a function of z, evaluation of the limit (24.1) is somewhat
easier. We find
f(z + ∆z) − f(z)
f (z) = lim
∆z→0
∆z
1
1
1
= lim
−
∆z→0 ∆z
1 − z − ∆z
1−z
1
1
=
= lim
,
∆z→0 (1 − z − ∆z)(1 − z)
(1 − z)2
independently of the way in which ∆z → 0, provided z = 1. Hence f(z) is analytic
everywhere except at the singularity z = 1. 24.2 The Cauchy–Riemann relations
From examining the previous examples, it is apparent that for a function f(z)
to be differentiable and hence analytic there must be some particular connection
between its real and imaginary parts u and v.
By considering a general function we next establish what this connection must
be. If the limit
f(z + ∆z) − f(z)
L = lim
(24.2)
∆z→0
∆z
is to exist and be unique, in the way required for differentiability, then any two
specific ways of letting ∆z → 0 must produce the same limit. In particular, moving
parallel to the real axis and moving parallel to the imaginary axis must do so.
This is certainly a necessary condition, although it may not be sufficient.
If we let f(z) = u(x, y) + iv(x, y) and ∆z = ∆x + i∆y then we have
f(z + ∆z) = u(x + ∆x, y + ∆y) + iv(x + ∆x, y + ∆y),
and the limit (24.2) is given by
u(x + ∆x, y + ∆y) + iv(x + ∆x, y + ∆y) − u(x, y) − iv(x, y)
.
L = lim
∆x, ∆y→0
∆x + i∆y
If we first suppose that ∆z is purely real, so that ∆y = 0, we obtain
u(x + ∆x, y) − u(x, y)
v(x + ∆x, y) − v(x, y)
∂v
∂u
+i
+i ,
L = lim
=
∆x→0
∆x
∆x
∂x
∂x
(24.3)
provided each limit exists at the point z. Similarly, if ∆z is taken as purely
imaginary, so that ∆x = 0, we find
u(x, y + ∆y) − u(x, y)
v(x, y + ∆y) − v(x, y)
∂v
1 ∂u
+i
+
.
L = lim
=
∆y→0
i∆y
i∆y
i ∂y ∂y
(24.4)
827
COMPLEX VARIABLES
For f to be differentiable at the point z,
be identical. It follows from equating real
conditions for this are
∂v
∂u
=
and
∂x
∂y
expressions (24.3) and (24.4) must
and imaginary parts that necessary
∂v
∂u
=− .
∂x
∂y
(24.5)
These two equations are known as the Cauchy–Riemann relations.
We can now see why for the earlier examples (i) f(z) = x2 − y 2 + i2xy might
be differentiable and (ii) f(z) = 2y + ix could not be.
(i) u = x2 − y 2 , v = 2xy:
∂v
∂u
= 2x =
∂x
∂y
∂v
∂u
= 2y = − ,
∂x
∂y
and
(ii) u = 2y, v = x:
∂u
∂v
=0=
∂x
∂y
but
∂v
∂u
= 1 = −2 = − .
∂x
∂y
It is apparent that for f(z) to be analytic something more than the existence
of the partial derivatives of u and v with respect to x and y is required; this
something is that they satisfy the Cauchy–Riemann relations.
We may enquire also as to the sufficient conditions for f(z) to be analytic in
R. It can be shown§ that a sufficient condition is that the four partial derivatives
exist, are continuous and satisfy the Cauchy–Riemann relations. It is the additional requirement of continuity that makes the difference between the necessary
conditions and the sufficient conditions.
In which domain(s) of the complex plane is f(z) = |x| − i|y| an analytic function?
Writing f = u + iv it is clear that both ∂u/∂y and ∂v/∂x are zero in all four quadrants
and hence that the second Cauchy–Riemann relation in (24.5) is satisfied everywhere.
Turning to the first Cauchy–Riemann relation, in the first quadrant (x > 0, y > 0) we
have f(z) = x − iy so that
∂u
∂v
= 1,
= −1,
∂x
∂y
which clearly violates the first relation in (24.5). Thus f(z) is not analytic in the first
quadrant.
Following a similiar argument for the other quadrants, we find
∂u
= −1 or
∂x
∂v
= −1 or
∂y
+1
for x < 0 and x > 0, respectively,
+1
for y > 0 and y < 0, respectively.
Therefore ∂u/∂x and ∂v/∂y are equal, and hence f(z) is analytic only in the second and
fourth quadrants. §
See, for example, any of the references given on page 824.
828
24.2 THE CAUCHY–RIEMANN RELATIONS
Since x and y are related to z and its complex conjugate z ∗ by
1
1
(z + z ∗ )
and
y = (z − z ∗ ),
(24.6)
2
2i
we may formally regard any function f = u + iv as a function of z and z ∗ , rather
than x and y. If we do this and examine ∂f/∂z ∗ we obtain
x=
∂f
∂f ∂x
∂f ∂y
=
+
∂z ∗
∂x ∂z ∗
∂y ∂z ∗
∂u
∂v
1
∂u
∂v
1
+i
+i
=
+
−
∂x
∂x
2
∂y
∂y
2i
∂v
∂u
1 ∂u
i ∂v
−
+
=
+
.
2 ∂x ∂y
2 ∂x ∂y
(24.7)
Now, if f is analytic then the Cauchy–Riemann relations (24.5) must be satisfied,
and these immediately give that ∂f/∂z ∗ is identically zero. Thus we conclude that
if f is analytic then f cannot be a function of z ∗ and any expression representing
an analytic function of z can contain x and y only in the combination x + iy, not
in the combination x − iy.
We conclude this section by discussing some properties of analytic functions
that are of great practical importance in theoretical physics. These can be obtained
simply from the requirement that the Cauchy–Riemann relations must be satisfied
by the real and imaginary parts of an analytic function.
The most important of these results can be obtained by differentiating the
first Cauchy–Riemann relation with respect to one independent variable, and the
second with respect to the other independent variable, to obtain the two chains
of equalities
∂ ∂v
∂ ∂v
∂ ∂u
∂ ∂u
=
=
=−
,
∂x ∂x
∂x ∂y
∂y ∂x
∂y ∂y
∂ ∂u
∂ ∂u
∂ ∂v
∂ ∂v
=−
=−
=−
.
∂x ∂x
∂x ∂y
∂y ∂x
∂y ∂y
Thus both u and v are separately solutions of Laplace’s equation in two dimensions, i.e.
∂2 u
∂2 u
+ 2 =0
2
∂x
∂y
and
∂2 v
∂2 v
+ 2 = 0.
2
∂x
∂y
(24.8)
We will make significant use of this result in the next chapter.
A further useful result concerns the two families of curves u(x, y) = constant
and v(x, y) = constant, where u and v are the real and imaginary parts of any
analytic function f = u + iv. As discussed in chapter 10, the vector normal to the
curve u(x, y) = constant is given by
∇u =
∂u
∂u
i+
j,
∂x
∂y
829
(24.9)