ENGG2420D
Notes for Lecture 6
Complex differentiability at a given point
Kenneth Shum
7/10/2015
Remarks on distance and circle in complex plane: Distance in complex number is measured by the modulus, or the absolute value. Two complex
numbers z1 and z2 are within a distance r if
|z1 − z2 | ≤ r.
Numerical example: The distance between 1 − i and 2 + 3i is
√
|1 − i − (2 + 3i)| = | − 1 − 4i| = 17.
Given a complex number z0 and a positive real number r, the complex numbers
which are within a distance of r from z0 form a circular disc. These are the
complex numbers satisfying the condition
|z − z0 | ≤ r.
Using notation from set theory, the disc with radius r and center z0 is the set
{z ∈ C : |z − z0 | ≤ r}.
Numerical example: The disc with radius 5 and center −i in the complex
plane consists of points in
{z ∈ C : |z + i| ≤ 5}.
If you wish to convert it to real and imaginary parts x and y, then it is
p
{x + iy ∈ C : x2 + (y − 1)2 ≤ 5}.
1
Complex differentiable at a point
A complex function f is said to be complex differentiable at a given point z0 if
there exists a complex constant L such that
f (z) − f (z0 )
→L
z − z0
1
as z → z0 .
(1)
In alternate notations, this means that the limit
lim
z→z0
f (z) − f (z0 )
z − z0
exists and is equal to L. In the above limit, the complex variable z may approach
the given point z0 in any direction. Intuitively, complex differentiable at z0
(z0 )
is
means that for all z which is sufficiently close to z0 , the fraction f (z)−f
z−z0
approximately equal to L. The closer to z0 , the smaller the error.
The constant L is called the complex derivative of f at z0 . We often denote
the limit L by f 0 (z0 ).
(In -δ formalism, when we say that f is complex differentiable at a given
point z0 , we mean that for any arbitrarily small > 0, we can a δ > 0 (which may
(z0 )
−L ≤ depend on ) such that for all z satisfying |z−z0 | ≤ δ, we have f (z)−f
z−z0
for some choice of complex number L.)
We let h be the difference between a generic complex number z and the given
complex number z0 . The definition in (1) is equivalent to
f (z0 + h) − f (z0 )
→L
h
as h → 0.
(2)
From (2), we have yet another formulation of complex differentiability at a
point z0
.
f (z0 + h) − f (z0 ) = Lh
for complex number h which is very close to the origin. In terms of the little o
notation, an alternate way to write this is
f (z0 + h) − f (z0 ) = L · h + o(h),
(3)
f (z) = f (z0 ) + L · (z − z0 ) + o(z − z0 ).
(4)
or
(We hide the process of taking limit by using the notation o(h), because by
definition, o(h) stands for any function satisfying limh→0 |o(h)/h| = 0. For
example for any integer m ≥ 2, the function hm decreases to 0 faster than h,
since limh→0 |hm /h| = limh→0 |hm−1 | = 0.)
We see that the original definition of complex differentiability in (1) is equivalent to the linear approximation in (4).
2
An extended example for f (z) = eaz
Question:
Is f (z) = eiz complex differentiable at z0 = −1?
We answer this question by providing an answer to a more generic question:
2
For any fixed complex numbers a and z0 , is f (z) = eaz complex
differentiable at z0 ?
In terms of linear approximation, this question is equivalent to
For any fixed complex numbers a and z0 , can we approximate f (z) =
eaz by
f (z) = f (z0 ) + L · (z − z0 )
when z is close to z0 , for some complex number L?
To answer this question, we recall from calculus that the real exponential
function ex can be approximated by the linear function
.
ex = 1 + x
when x is small.
y = ex
y
y =1+x
x
We want to extend this fact to the complex exponential function. Suppose
h = x + iy is a complex number with very small x and y. By definition,
eh = ex+iy = ex eiy = ex (cos y + i sin y).
We use the fact that when y is small, cos y is close to 1 and sin y is close to y,
and get
.
eh = (1 + x)(1 + iy) = 1 + x + iy + smaller order term = 1 + h + o(h).
Therefore,
f (z0 + h) = ea(z0 +h)
= eaz0 eah
= eaz0 (1 + ah + o(h))
= eaz0 + aez0 · h + o(h).
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We have the linear approximation
eaz = eaz0 + aeaz0 (z − z0 ) + o(z − z0 )
for z which is sufficiently close to z0 .
In particular, when a = i and z0 = −1, we have the linear approximation
.
eiz = e−i + ie−i (z + 1)
for z which is sufficiently close to −1.
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Relationship to angle-preserving property
Suppose that f is complex differentiable at z0 , and suppose that
f 0 (z0 ) 6= 0.
Consider two small complex numbers h1 and h2 with very small absolute values,
representing two different perturbations from z0 . Let the angle between the line
segment from z0 to z0 + h1 and the line segment from z0 to z0 + h2 be α, i.e.,
(z + h ) − z h 0
2
0
2
α = arg
= arg
.
(z0 + h1 ) − z0
h1
Now we turn our attention to the image of z0 , z0 + h1 and z0 + h2 in the range of
f , namely f (z0 ), f (z0 + h1 ) and f (z0 + h2 ). What is the angle between the line
segment from f (z0 ) to f (z0 + h1 ) and the line segment from f (z0 ) to f (z0 + h2 )?
From the linear approximation of f near z0 , we get
.
f (z0 + h1 ) − f (z0 ) = f 0 (z0 )h1
.
f (z0 + h2 ) − f (z0 ) = f 0 (z0 )h2 .
Hence,
arg
f ((z + h )) − f (z ) f 0 (z )h .
0
2
0
0 2
= arg 0
f ((z0 + h1 )) − f (z0 )
f (z0 )h1
Since f 0 (z0 ) is non-zero by assumption, the angle is equal to arg(h2 /h1 ) = α.
This shows that the angle is preserved locally in a small region around z0 .
z0 + h2
f (z0 + h1 )
f (z0 + h2 )
f
h2
z0 + h1
h1
z0
f (z0 )
4
Summary
1. A complex function f (z) is complex differentiable at a given point z0
precisely when f (z) can be approximately by a complex linear function
of the form a + b(z − z0 ) for z near z0 . The constant b in the linear
approximation is the complex derivative of f (z) at z0 .
2. When f is complex differentiable at z0 and the derivative f 0 (z0 ) is nonzero, we have the angle-preserving property at z = z0 .
3. For any fixed complex number a, the function eaz is complex differentiable
at any point z0 , and the derivative is aeaz0 .
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