Complex Arithmetic
The Real Numbers
We assume the reader to be familiar with the real number systemR We let
R 2 = {(x, y) : x ∈R , y ∈ R} (Figure 1.1) These are ordered pairs of real numbers.
As we shall see, the complex numbers are nothing other thanR 2 equipped with a special algebraic structure.
The Complex Numbers
The complex numbers C consist of R 2 equipped with some binary algebraic operations One defines
These operations of + andã are commutative and associative.
We denote (1,0) by 1 and (0,1) by i If α ∈ R, then we identify α with the complex number (α,0) Using this notation, we see that αã(x, y) = (α,0)ã(x, y) = (αx, αy) (1.1.2.1)
As a result, if (x, y) isany complex number, then
(x, y) = (x,0) + (0, y) =xã(1,0) +yã(0,1) =xã1 +yãi≡x+iy
Every complex number can be uniquely expressed in the form x + iy, where x and y are real numbers This notation simplifies the representation of complex numbers, allowing for clearer mathematical operations The fundamental laws of addition and multiplication apply to these numbers, facilitating their use in various mathematical contexts.
(x+iy)ã(x 0 +iy 0 ) = (xx 0 −yy 0 ) +i(xy 0 +yx 0 ).
Observe thatiãi=−1.Finally, the multiplication law is consistent with the scalar multiplication introduced in line (1.1.2.1).
In the realm of complex numbers, the symbols z, w, and ζ represent complex values, typically expressed as z = x + iy, w = u + iv, and ζ = ξ + iη Here, the real part of z is denoted as x, written as Re(z), while the imaginary part is represented by y, noted as Im(z).
The complex conjugate of a complex number is defined as x - iy, where x + iy represents the original complex number Denoted as z, the conjugate is expressed as z = x - iy This concept is particularly important because if p is a quadratic polynomial with real coefficients and z is one of its roots, then the conjugate z is also a root of the polynomial.
Figure 1.2: Euclidean distance (modulus) in the plane.
Complex Conjugate
Note that z+z = 2x , z−z = 2iy Also z+w=z+w , (1.1.3.1) zãw=zãw (1.1.3.2)
A complex number is real (has no imaginary part) if and only ifz =z.It is imaginary (has no real part) if and only ifz =−z.
Modulus of a Complex Number
The ordinary Euclidean distance of (x, y) to (0,0) is p x 2 +y 2 (Figure 1.2).
We also call this number themodulusof the complex numberz =x+iy and we write|z|=p x 2 +y 2 Note that zãz =x 2 +y 2 =|z| 2
The distance fromztowis|z−w|.We also have the formulas|zãw|=|z|ã|w| and|Rez| ≤ |z| and |Imz| ≤ |z|.
The Topology of the Complex Plane
IfP is a complex number and r >0, then we set
Figure 1.3: Open and closed discs. and
The first type of disc is the open disc centered at point P with radius r, while the second type is the closed disc with the same center and radius For convenience, we commonly use the symbols D and D to represent the open disc D(0,1) and the closed disc D(0,1), respectively.
A subset U of the complex numbers C is defined as open if, for every point P in C, there exists a radius r > 0 such that the disk D(P, r) is entirely contained within U This means that each point P in the set has neighboring points within a distance less than r that also belong to the set For instance, the set U = {z ∈ C: Re(z) > 1} is open, whereas the set F = {z ∈ C: Re(z) ≤ 1} is not.
A set E ⊆ C is said to be closed if C\E ≡ {z ∈ C : z 6∈ E} (the complement of E inC) is open The setF in the last paragraph is closed.
It isnotthe case that any given set is either open or closed For example, the setW ={z ∈C: 10 be such that D(P, r)⊆U Let γ : [0,1]→Cbe theC 1 curve γ(t) =P +rcos(2πt) +irsin(2πt) Then, for eachz ∈D(P, r), f(z) = 1
One may derive this result directly from Stokes’s theorem (see [KRA5] and also our Subsection 2.3.1).
More General Forms of the Cauchy Theorems
Now we present the very useful general statements of the Cauchy integral theorem and formula First we need a piece of terminology A curve γ : [a, b]→C is said to be piecewiseC k if
[a, b] = [a0, a1]∪[a1, a2]∪ ã ã ã ∪[am − 1, am] (2.3.3.1) with a =a0 < a1 0 We apply the Cauchy estimate (3.1.2.1) for k = 1 on
Since this inequality is true for everyr >0,we conclude that
Since P was arbitrary, we conclude that
The end of the last proof bears some commentary We prove that∂f/∂z ≡
0 But we know, since f is holomorphic, that ∂f/∂z ≡ 0 It follows from linear algebra that∂f/∂x≡0 and ∂f/∂y≡0 Then calculus tells us that f is constant.
Liouville's theorem can be applied to demonstrate a broader principle: for an entire function f: C → C, if there exists a real number C and a positive integer k such that certain conditions are met, then specific conclusions about the function can be drawn.
|f(z)| ≤Cã(1 +|z|) k for allz, thenf is a polynomial in z of degree at most k.
3.1 The Derivatives of a Holomorphic Function 41
The Fundamental Theorem of Algebra
One of the most elegant applications of Liouville’s Theorem is a proof of what is known as the Fundamental Theorem of Algebra (see also§§1.1.7):
The Fundamental Theorem of Algebra: Letp(z) be a non- constant (holomorphic) polynomial Then p has a root That is, there exists an α∈C such that p(α) = 0.
Proof: Suppose not Then g(z) = 1/p(z) is entire Also when |z| → ∞, then |p(z)| → +∞ Thus 1/|p(z)| → 0 as |z| → ∞; hence g is bounded.
By Liouville’s Theorem, g is constant, hence p is constant Contradiction.
For a polynomial \( p \) of degree \( k \geq 1 \), the Fundamental Theorem guarantees the existence of a root \( \alpha_1 \) Utilizing the Euclidean algorithm, we can divide \( z - \alpha_1 \) into \( p \) without any remainder, resulting in the expression \( p(z) = (z - \alpha_1) \tilde{p_1}(z) \).
A polynomial \( p_1 \) of degree \( k-1 \) has a root \( \alpha_2 \) if \( k-1 \geq 1 \), making it divisible by \( (z - \alpha_2) \) This leads to the expression \( p(z) = (z - \alpha_1)^{a}(z - \alpha_2)^{a} p_2(z) \), where \( p_2(z) \) is a polynomial of degree \( k - 2 \) This factorization process continues until reaching a constant polynomial \( p_k \) of degree 0 Consequently, if \( p(z) \) is a holomorphic polynomial of degree \( k \), it can be expressed in the form \( p(z) = C (z - \alpha_1)^{a} (z - \alpha_2)^{a} \cdots (z - \alpha_k)^{a} \), where \( C \) is a non-zero constant and \( \alpha_1, \ldots, \alpha_k \) are complex numbers that may not be distinct.
If some of the roots of p coincide, then we say that p has multiple roots.
A polynomial \( p \) is said to have a root of order \( m \) at a complex number \( \alpha \) if \( m \) of its values \( \alpha_j \) are equal to \( \alpha \) This implies that \( p(\alpha) = 0 \) and all derivatives up to order \( m-1 \) at \( \alpha \) also equal zero, indicating the root's multiplicity.
An example will make the idea clear: Let p(z) = (z−5) 3 ã(z+ 2) 8 ã(z−3i)ã(z+ 6).
The polynomial p exhibits a root of order 3 at the value 5, an order 8 root at -2, and simple roots of order 1 at 3i and -6 Additionally, p has simple roots at both 1 and -6.
Sequences of Holomorphic Functions and their Deriva-
A sequence of functions \( g_j \) defined on a common domain \( E \) converges uniformly to a limit function \( g \) if, for every \( \epsilon > 0 \), there exists a number \( N > 0 \) such that for all \( j > N \), the inequality \( |g_j(x) - g(x)| < \epsilon \) holds for every \( x \in E \) This means that the proximity of \( g_j(x) \) to \( g(x) \) does not depend on the choice of \( x \) within the domain \( E \).
Let \( f_j : U \rightarrow \mathbb{C} \) (for \( j = 1, 2, 3, \ldots \)) be a sequence of holomorphic functions defined on an open set \( U \) in \( \mathbb{C} \) If there exists a function \( f : U \rightarrow \mathbb{C} \) such that the sequence \( f_j \) converges uniformly to \( f \) on every compact subset \( E \) of \( U \), then \( f \) is holomorphic on \( U \) Consequently, \( f \) is infinitely differentiable, meaning \( f \in C^\infty(U) \).
Iffj, f, U are as in the preceding paragraph, then, for anyk ∈ {0,1,2, }, we have
3.1 The Derivatives of a Holomorphic Function 43 uniformly on compact sets 1 The proof is immediate from (3.1.1.1), which we derived from the Cauchy integral formula, for the derivative of a holomorphic function.
The Power Series Representation of a Holomorphic Func-
The ideas being considered in this section can be used to develop our under- standing of power series A power series
X∞ j=0 aj(z−P) j (3.1.6.1) is defined to be the limit of its partial sums
We say that the partial sumsconvergeto the sum of the entire series. Any given power series has adisc of convergence More precisely, let r= 1 lim sup j →∞ |aj| 1/j (3.1.6.3)
The power series (3.1.6.2) converges within the disc D(P, r), exhibiting absolute and uniform convergence on any smaller disc D(P, r0) where r0 < r It is important to note that in numerous examples, the sequence demonstrates this behavior.
The sequence |aj|^(1/j) converges as j approaches infinity, allowing us to define r as 1/lim(j→∞)|aj|^(1/j) However, if the sequence {|aj|^(1/j)} does not converge, it is necessary to refer to the formal definition of r For further details, consult sources [KRA3] and [RUD1].
The partial sums of the power series are polynomials that are holomorphic within any disc D(P, r) If the power series converges in the disc D(P, r), we denote the resulting function as f, which represents the limit of the series.
1 It is also common to say that the functions converge normally. uniformly on D(P, r 0 ) We can conclude immediately that f(z) is holomor- phic onD(P, r) Moreover, we know that
A differentiated power series maintains a disc of convergence that is at least as large as that of the original series, centered at the same point Within this disc, the differentiated power series converges to the derivative of the sum of the original series.
In complex function theory, a key aspect of power series is that if \( f \) is a holomorphic function defined on a domain \( U \subseteq \mathbb{C} \), and \( P \) is a point within \( U \) such that the disk \( D(P, r) \) is entirely contained in \( U \), then \( f \) can be expressed as a convergent power series within that disk Specifically, this is represented by the formula \( f(z) = \sum_{j=0}^{\infty} a_j(z - P)^j \), where the coefficients \( a_j \) are determined by \( a_j = \frac{f^{(j)}(P)}{j!} \).
The formula presented allows for the explicit calculation of the power series expansion of any holomorphic function f around a point P within its domain Additionally, it provides a priori knowledge regarding the convergence disc of the power series representation.
The topic warrants deeper exploration, particularly regarding Taylor series expansions of smooth functions While every smooth function \( f(x) \) can be expressed as a Taylor series around any point \( p \) within its domain, this series often fails to converge, and even when it does, it may not equal \( f \) In contrast, holomorphic functions of a complex variable exhibit a significant difference: their Taylor or power series expansions consistently converge The proof of this is straightforward, as demonstrated by applying the Cauchy formula with the center of the disc set at the origin.
The Zeros of a Holomorphic Function
The Zero Set of a Holomorphic Function
If \( f \) is a holomorphic function that is not identically zero, it cannot have too many zeros, reflecting the principle that holomorphic functions share characteristics with polynomials To articulate this idea clearly, we must consider the topological concept of connectedness.
Discreteness of the Zeros of a Holomorphic Function
Let U ⊆C be a connected (§§1.1.5) open set and let f :U →C be holomorphic Let the zero set of f beZ ={z ∈U :f(z) = 0}.
If there are a z0 ∈ Z and {zj} ∞ j=1 ⊆ Z \ {z0} such that zj →z0, then f ≡0.
In topological terms, a point \( z_0 \) is considered an accumulation point of a set \( Z \) if there exists a sequence \( \{z_j\} \subset Z \setminus \{z_0\} \) such that \( \lim_{j \to \infty} z_j = z_0 \) The theorem can be restated as follows: if \( f: U \to \mathbb{C} \) is a holomorphic function defined on a connected open set \( U \), and if the set \( Z = \{ z \in U : f(z) = 0 \} \) contains an accumulation point in \( U \), then it follows that \( f \equiv 0 \).
If the point 0 is an interior accumulation point of the zeros {zj} of the holomorphic function f, then f(0) must equal 0 This allows us to express f(z) as z^a f*(z), where f* also vanishes at {zj}, indicating that 0 remains an accumulation point of these zeros Consequently, we find that f*(0) = 0, leading to the conclusion that f has a zero of order 2 at 0 By continuing this process, we can determine that f has a zero of infinite order at 0, resulting in the power series expansion of f around 0 being identically 0 This implies, through a straightforward connectedness argument, that f is identically equal to 0.
Discrete Sets and Zero Sets
In the study of holomorphic functions, a set is defined as discrete if each point in the set has a neighborhood that contains no other points from the set This concept leads to the understanding that the zero set of a non-constant holomorphic function on a connected open set consists of isolated points However, it is crucial to note that this does not exclude the possibility of accumulation points existing outside the open set For example, the function \( f(z) = \sin(1/[1-z]) \) demonstrates that zeros can accumulate at the boundary of the unit disc, specifically at the point 1.
3.2 The Zeros of a Holomorphic Function 47
Figure 3.3: Zeros accumulating at a boundary point.
Uniqueness of Analytic Continuation
A consequence of the preceding basic fact (§§3.2.2) about the zeros of a holomorphic function is this: Let U ⊆ C be a connected open set and D(P, r) ⊆ U If f is holomorphic on U and f
U In fact if f ≡0 on a segment then it must follows that f ≡0.
Here are some further corollaries:
(3.2.4.1) Let U ⊆ C be a connected open set Let f, g be holomorphic on
U If {z∈U :f(z) =g(z)} has an accumulation point inU, then f ≡g. (3.2.4.2)LetU ⊆Cbe a connected open set and letf, g be holomorphic on
U If fãg ≡0 on U, then eitherf ≡0 on U or g ≡0 on U.
(3.2.4.3) Let U ⊆ C be connected and open and let f be holomorphic on
U If there is a P ∈U such that
If f and g are entire holomorphic functions such that f(x) = g(x) for all real x, then it follows that f is identically equal to g Additionally, functional identities that hold for all real values also extend to complex values For example, the identity sin²z + cos²z = 1 is valid for all z in the complex plane, as it is true for all real numbers x.
“principle of persistence of functional relations”—see [GRK].
3.2 The Zeros of a Holomorphic Function 49
Figure 3.4: Principle of persistence of functional relations.
The Behavior of a Holomorphic Function near an Isolated Sin-
Isolated Singularities
It is often important to consider a function that is holomorphic on a punc- tured open setU \ {P} ⊂C Refer to Figure 4.1.
This chapter introduces a novel infinite series expansion that extends the concept of power series expansion for holomorphic functions around nonsingular points Additionally, it provides a comprehensive classification of the behavior of holomorphic functions in the vicinity of isolated singular points.
A Holomorphic Function on a Punctured Domain
In the context of complex analysis, let U be an open set in the complex plane, and let P be a point within U The set U \ {P} is referred to as a punctured domain When a function f is holomorphic on this punctured domain, we describe P as an isolated singular point of f This terminology indicates that f remains defined and holomorphic in a neighborhood surrounding P, excluding the point itself While the specific characteristics of the open set U are of secondary importance, our primary focus lies in analyzing the behavior of the function f near the isolated singularity at P.
Classification of Singularities
There are three possibilities for the behavior of f near P that are worth distinguishing:
(4.1.3.1) |f(z)| is bounded onD(P, r)\ {P}for some r >0 with D(P, r) ⊆
U; i.e., there is some r > 0 and some M > 0 such that |f(z)| ≤ M for all z ∈U ∩D(P, r)\ {P}.
Elementary logic indicates that these three conditions encompass all possibilities Although the description in (4.1.3.3) may seem unsatisfactory, it highlights a complex situation that lacks a straightforward explanation We will delve deeper into each of these three conditions in the following discussion.
4.1 Behavior Near an Isolated Singularity 53
Removable Singularities, Poles, and Essential Singu-
If condition (4.1.3.1) is satisfied, then the function f has a limit at point P, allowing it to be extended and become holomorphic throughout the region U, which is a result established by Riemann In this scenario, f is described as having a removable singularity at P Conversely, if condition (4.1.3.2) applies, f is characterized as having a pole at P Lastly, in the case of condition (4.1.3.3), f is said to exhibit an essential singularity at P.
P.Our goal in this and the next two subsections is to understand (4.1.3.1)–(4.1.3.3)in some further detail.
The Riemann Removable Singularities Theorem
Letf :D(P, r)\ {P} →Cbe holomorphic and bounded Then
(4.1.5.2) The function fb:D(P, r)→C defined by fb(z) ( f(z) if z 6=P ζlim→ Pf(ζ) if z =P is holomorphic.
To prove the statement, let P = 0 and define the function g(z) = z²f(z) It can be verified that g is continuously differentiable (C¹) and satisfies the Cauchy-Riemann equations throughout the disk D(P, r) The boundedness condition ensures that both g and its first derivative approach a limit at 0 Consequently, g is holomorphic within the disk and has a zero of second order at 0 Therefore, the function f(z) = g(z)/z² is also a well-defined holomorphic function across the entire disk D(P, r).
The Casorati-Weierstrass Theorem
If f : D(P, r0)\ {P} → C is holomorphic and P is an essential singularity of f, then f(D(P, r) \ {P}) is dense in C for any
Assuming the assertion is false, we find a complex value and a positive number such that the image of D(P, r) \ {P} under the function f does not include the disc D(à, ) Consequently, the function g(z) = 1/[f(z)−à] remains bounded and non-vanishing near P, indicating that it possesses a removable singularity This leads us to conclude that f is bounded near P, which contradicts our initial assumption.
At a removable singularity P, a holomorphic function f defined on D(P, r0) \ {P} can be extended to be holomorphic on the entire domain D(P, r0) In contrast, near an essential singularity at P, a holomorphic function g on D(P, r0) \ {P} exhibits a dense image in the complex plane C The third scenario, where a holomorphic function h has a pole at P, will be explored in greater detail in the following sections.
We next develop a new type of doubly infinite series that will serve as a tool for understanding isolated singularities—especially poles.
Expansion around Singular Points
Laurent Series
A Laurent series on D(P, r) is a (formal) expression of the form
Note that the individual terms are each defined for all z ∈ D(P, r)\ {P}.The series sums fromj =−∞ toj = +∞.
Convergence of a Doubly Infinite Series
To discuss convergence of Laurent series, we must first make a general agree- ment as to the meaning of the convergence of a “doubly infinite” series
P+ ∞ j= −∞αj We say that such a series converges if P+ ∞ j=0αj and P+ ∞ j=1α − j P − 1 j= −∞αj converge in the usual sense In this case, we set
In other words, the question of convergence for a bi-infinite series devolves to two separate questions about two sub-series.
We can now present the analogues for Laurent series of our basic results for power series.
Annulus of Convergence
The convergence set of a Laurent series can be described as an open set of the form {z: 0≤r1