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9
Complex-valued Functions
The origin of the concept of “function”
A function is a relation that uniquely
associates members of one set with members of another set. More formally, a function from
A to B is an object f such that every a in A is uniquely associated with an object f (a) in
B.
As a mathematical term, “function” was coined by Gottfried Leibniz, in a 1673 letter, to
describe a quantity related to a curve, such as a curve’s slope at a specific point. In fact, the
functions Leibniz considered are today called differentiable functions, which are the basis of
calculus.
In the Introductio Euler made the function the central concept of analysis. His definition
of a function is essentially the one we use today in applied mathematics and physics (although
in pure mathematics it has been replaced by the “mapping” concept):
“A function of a variable quantity is any analytic expression whatsoever made
up from that variable quantity and from numbers or constant quantities.” 31
Euler’s teacher, Johann Bernoulli, defined it in terms very similar to Euler’s. But it
was Euler who introduced the modern notation f (x) for a function and used it for all kinds
of functions-explicit and implicit, continuous and discontinuous, and functions of several
independent variables. Euler also made free use of the expansion of functions in infinite
series and products - often with a carefree attitude that would not be tolerated today.
Complex-valued functions We call a function f : Ω → C, where Ω is a subset of C, a
complex-valued function. There are three functions associated with f :
f (z) := f (z), Re(f ) =
f −f
f +f
, Im(f ) =
.
2
2i
[Example]
f = z 2 + 3zz + 2z 2 is a complex-valued function. If we use the formulas
z = x + iy and z = x − iy, the corresponding function is
f = (x + iy)2 + 3(x + iy)(x − iy) + 2(x − iy)2
= (x2 − y 2 + 2ixy) + 3(x2 + y 2) + 2x2 − 2y 2 − 4ixy = 6x2 − 2ixy. Essentially, a complex-valued function is a map from R2 to R2 . In this examle, it is (x, y) 7→
(6x2 , −2xy).
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e: The story of a number, by Eli Maor, Princeton Press, 1994, p.168.
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[Example]
If a complex-valued function f has power series, f can be written as
f (z) =
∞
X
aij z i z j .
i,j=0
P
P∞
P∞
i j
i j
i j
Recall f is real-valued if and onl if f = f , i.e., ∞
i,j=0 aij z z =
i,j=0 aij z z =
i,j aji z z ,
i.e.,
aij = aji , ∀i, j ≥ 0.
Sometimes, we need to use real-valued functions, for example, a domain in C can be defined
as Ω = {z | ρ(z, z) < 0} where ρ(z, z) is a real-valued function. Then its boundary ∂Ω =
{z | ρ(z, z) = 0}.
For example, a unit circle ∂∆(1) = {x2 + y 2 − 1 = 0}. Recall x =
z+z
2
and y =
z−z
.
2i
Then
2
z+z 2
z−z
∂∆ = z +
− 1 = 0 = {z | zz − 1 = 0} = {z | |z| = 1}. 2
2i
[Example]
Let us consider two polynomials. First: f = zz − 1, i.e., f = |z|2 − 1. Its
zero set {f = 0} = {z | |z|2 − 1 = 0} is a circle. Second: g = az 2 + bz + c. Its zero set
{g = 0} = {z | az 2 + bz + c = 0} has exactly two complex roots.
In general, for a complex-valued function f , its zero set {f = 0} may not be isolated
points. However,
if we consider a function which does not contain any term involving z j ,
P
i.e., f (z) = j aj z j , then the set {f = 0} must be a set of isolated points (It’ll be proved
late). A such function will be called a holomorphic function, which is the main object to be
studied.
Limit of a complex-valued function
Let f : Ω → C be a complex-valued function.
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Let a be a limit point of Ω.
We say that f has limit A ∈ C at a if ∀ǫ > 0, ∃δ > 0 such
that
|f (z) − A| < ǫ, ∀z ∈ ∆(a, δ) − {a} ∩ Ω.
In this case, we denote limz→a f (z) = A.
A complex-valued function f defined in A is called continuous at z0 ∈ A if
lim f (z) = f (z0 ).
z→z0
[Example] We can use the definition to verify a limit by the same way in real analysis.
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Notice, by the definition, that a is not necessary in Ω.
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For example, let us prove:
lim (5z − 4) = 6.
z→2
Proof: For any ǫ > 0, we want to find a number δ > 0 such that whenever 0 < |z − 2| < δ,
we should have
|(5z − 4) − 6| < ǫ.
(35)
Since |(5z − 4) − 6)| = |5z − 10| = 5|z − 2| ≤ 5δ, we can take δ so that 5δ < ǫ, i.e., we can
take δ := 5ǫ , such that (35) is true. f is called bounded on Ω if ∃M > 0 such that
|f (z)| ≤ M, ∀z ∈ Ω.
Theorem 9.1 Let A ⊂ C be compact and f : A → C be a continuous complex-valued
function. Then
(i) f is bounded and |f (z)| attains the maximum. 33
(ii) f is uniformly continuous, i.e., ∀ǫ > 0, ∃δ > 0 such that
|f (z) − f (w)| < ǫ,
Proof:
∀z, w ∈ A with |z − w| < δ.
(i) ∀a ∈ A, ∃∆(a, δa ) such that
|f (z) − f (a)| < 1,
∀z ∈ ∆(a, δa ) (since f is continuous, we take ǫ = 1)
which implies
|f (z)| ≤ 1 + |f (a)|, ∀z ∈ ∆(a, δa ).
=⇒ We have an open covering {∆(a, δa )}a∈A of A.
∵ A is compact,
∴ ∃ a finite subcovering of A, say,
∆(a1 , δ1 ) ∪ ... ∪ ∆(an , δn ) ⊃ A.
=⇒ ∀z ∈ A,
|f (z)| ≤ max1≤j≤n 1 + |f (aj )| .
In fact, ∀z ∈ A =⇒ z ∈ ∆(aj , δj ) for some j so that |f (z)| ≤ 1 + |f (aj )|. This proves the
boundedness.
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Since we cannot compare big or less for complex numbers, we cannot talk about the “maximum” for
the complex-valued f (z) but it makes sense for the real-valued function |f (z)|.
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Since |f | can attain its maximum value on each disk ∆(aj , δj ), |f | can attain its maximum
value on A.
(ii) The proof is similar to the proof of Theorem 5.1.
Sequences of Complex Valued Functions Let A ⊂ C and {fn (z)}∞
n=1 be a sequence of
complex-valued functions defined on a set A ⊂ C. We say that {fn } is convergent if ∀z ∈ A,
limn→∞ fn (z) exists. In this case, we denote f (z) = limn→∞ fn (z), or f = lim fn , or fn → f .
We say that {fn } uniformaly converges to a function f (z) on A if ∀ǫ > 0, ∃n0 ∈ N such that
|fn (z) − f (z)| < ǫ,
∀n ≥ n0 , ∀z ∈ A.
unif
In this case, we denote fn −−→ f on A.
unif
Theorem 9.2 (i) (Cauchy criterion) fn −−→ f on A if and only if ∀ǫ > 0, ∃n0 ∈ N such
that
|fm (z) − fn (z)| < ǫ, ∀m, n ≥ n0 , ∀z ∈ A.
unif
(ii) If fn are continuous and fn −−→ f on A, then f is continuous on A.
In 1821 A. L. Cauchy published a wrong theorem: The pointwise limit of a sequence
of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found
counterexamples in the context of Fourier series. Dirichlet then analyzed Cauchy’s proof
and found the mistake: the notion of pointwise convergence had to be replaced by uniform
convergence.
The concept of uniform convergence was probably first used by Christoph Gudermann,
who was the teacher of Karl Weierstrass, in 1838 where he used the phrase “convergence
in a uniform way.” Under the influence of Weierstrass and Bernhard Riemann this concept
and related questions were intensely studied at the end of the 19th century.
[Example] Let fn (x) = xn be continous defined on [0, 1]. Then its limit function f = lim fn
becomes
(
0, if x ∈ [0, 1);
(36)
f (x) :=
1, if x = 1,
which is not continuous.
[Example] Let f (x) =
Proof:
1
x
defined on (0, 1). Prove that f is not uniformly continuous.
For any x, y ∈ (0, 1), we have
1 1 |y − x|
|f (x) − f (y)| = − =
.
x y
|xy|
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Take ǫ := 1, for any 0 < δ < 1, we take x = 2δ , y = δ. Then |x − y| =
|f (x) − f (y)| =
δ
2
< δ, but
1
2 1
− = > ǫ.
δ δ
δ
Therefore, the function is not uniformly continuous on (0, 1).
Ascoli-Arzela Theorem It is known (see Theorem 9.1) that any bounded sequence of
real (or complex) numbers has a convergent subsequence. In the theory of functions, one
may ask a similar question: For a sequence of bounded continuous functions, is there any
uniformly convergent subsequence (so that its limit function is also continuous) ?
This is not true in general (see the example in (36)). However, by adding appropriate
conditions, this can be true.
Theorem 9.3 (Ascoli-Arzela) Let A ⊂ C be a compact subset, and fn functions defined on
A that are uniformly bounded and equicontinuous. Then {fn } has a uniformly convergent
subsequence.
Here we explain the conditions used in the above theorem:
(1) A sequence of complex-valued functions {fn } on A is called uniformly bounded if
∃M > 0 such that
|fn (z)| ≤ M, ∀z ∈ A, ∀n.
(2) A family34 F of complex-valued functions on A is called equicontinuous if ∀ǫ > 0,
∃δ > 0 such that
|f (z) − f (w)| < ǫ, ∀z, w ∈ A with |z − w| < δ, ∀ f ∈ F .
We skip its proof of the Ascoli-Arzela theorem which is standard in real analysis.
Notice the set A in Ascoli-Arzela theorem is required to be compact (e.g. [a, b] × [c, d]).
If A is an open set, we still can have similar result but we hvae to modify the condition (the
proof is a simple consequence of the Ascoli-Arzela theorem).
Let D ⊂ C be an open subset, and {fn } defined on D. We say that {fn } converges
uniformly on compact subsets on D if for any compact subset A ⊂⊂ D, the restriction {fn }
converges uniformly.
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countable or uncountable
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Theorem 9.4 (i) If fn are continuous on an open subset D ⊂ C, and {fn }∞
n=1 converges
uniformly on compact subsets of D, then f := limn fn is continous on D.
(ii) {fn }∞
n=1 converges uniformly on compact subsets of D if and only if ∀a ∈ D, ∃r > 0
such that {fn }∞
n=1 converges uniformly on ∆(a; r) ⊂ D.
Theorem 9.5 If {fn }∞
n=1 on D ⊂ C is uniformly bounded and equicontinuous on compact
subsets of D (i.e., they hold on any fixed relatively compact subset of D), then {fn }∞
n=1 has
a subsequence that converges uniformly on compact subsets of D.
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