Complex-Valued Functions: Domain, Range, Classification, and Operations

Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost, – W.S. Anglin.

Recall

The standard formula for finding the $n$-th roots of a complex number $w$ is $\boxed{z_k = \sqrt[n]{|w|} \cdot e^{i\left(\frac{\text{Arg}(w) + 2\pi k}{n}\right)}, \quad k = 0, 1, \ldots, n-1}$

Definition. The set of all nth roots of a complex number w, denoted $w^{\frac{1}{n}}$, consists of n distinct values. It is given by: $w^{\frac{1}{n}}$ = {$\sqrt[n]{w}, w_n\sqrt[n]{w}, w_n²\sqrt[n]{w}, ..., w_n^{n-1}\sqrt[n]{w}$} where:

• |w| is the magnitude (modulus) and $\text{Arg}(w)$ is the argument (angle) of w, $\text{Arg}(w) ∈ (−π,π].$.
• $w_n=ζ_n=e^{\frac{2πi}{n}}$ is a primitive nth root of unity.
• The index k ranges over integers from 0 to n − 1. Using k = 0, 1, … , n−1 ensures exactly n distinct roots.
• The starting point is $\sqrt[n]{w}$. You find the very first root (the principal root) by taking the $n$-th root of the length ($|w|$) and dividing the original angle by $n$.
• The term $w_n = e^{i \frac{2\pi}{n}}$ is a pure rotation. Multiplying by $w_n$ rotates a number by $\frac{2\pi}{n}$ radians (which is $\frac{360^\circ}{n}$). In other words, these roots are obtained by adding multiples of $\frac{2\pi}{n}$ to the argument of the principal root. Each root corresponds to rotating the principal nth root by $\frac{2\pi}{n}$ radians around the origin. This is why the roots form a regular polygon.

Exercises

• Given z = $(1+i\sqrt{3}),\text{ compute } z^{\frac{1}{4}}$

1. Polar Form Conversion. |z| = $\sqrt{1² +(\sqrt{3})²} = \sqrt{4} = 2, Arg(z) = arctan(\frac{\sqrt{3}}{1}) = arctan(\sqrt{3}) = \frac{\pi}{3}$. Thus, the polar for of z is $z = 2e^{\frac{\pi}{3}i}$

2. Calculating the Principal Fourth Root: To find $z^{\frac{1}{n}}$, we use De Moivre’s Theorem, which states that for any complex number z = re ^iθ, its n-th roots are given by: $z^{\frac{1}{n}} = r^{\frac{1}{n}}e^{i(\frac{θ+2\pi k}{n})}$, k = 0, 1, 2, …, n − 1.
   For the principal (or primary) fourth root (k = 0, n = 4): $z^{\frac{1}{4}} = (2e^{\frac{\pi}{3}i})^{\frac{1}{4}} = 2^{\frac{1}{4}}e^{\frac{\pi}{3\cdot 4}i}= \sqrt[4]{2}e^{\frac{\pi}{12}i}$

3. Finding All Fourth Roots. $z^{\frac{1}{n}} = r^{\frac{1}{n}}e^{i(\frac{θ+2\pi k}{n})} = r^{\frac{1}{n}}e^{i(\frac{θ}{n})}e^{i(\frac{2\pi k}{n})}$. $w_n = e^{i(\frac{2\pi}{n})}, \sqrt[n]{w} = \sqrt[n]{|w|}e^{i\frac{Arg(w)}{n}} = \sqrt[4]{2}e^{\frac{\pi}{12}i}$.

To find all four fourth roots of z, $w^{\frac{1}{n}} = ${$r^{\frac{1}{n}}\sqrt[n]{w}, r^{\frac{1}{n}}w_n\sqrt[n]{w}, r^{\frac{1}{n}}w_n²\sqrt[n]{w}, ..., r^{\frac{1}{n}}w_n^{n-1}\sqrt[n]{w}$} =[ n = 4] {$r^{\frac{1}{4}}\sqrt[4]{w}, r^{\frac{1}{4}}w_4\sqrt[4]{w}, r^{\frac{1}{4}}w_n²\sqrt[4]{w}, r^{\frac{1}{4}}w_n^{3}\sqrt[4]{w}$}

n = 4, $w_4 = i = e^{\frac{\pi}{2}i} = e^{\frac{6\pi}{12}i}$. Therefore, the four fourth roots of z are: $z^{\frac{1}{4}} =$ {$\sqrt[4]{2}e^{\frac{\pi}{12}i}, \sqrt[4]{2}e^{\frac{7\pi}{12}i}, \sqrt[4]{2}e^{\frac{13\pi}{12}i}, \sqrt[4]{2}e^{\frac{19\pi}{12}i}$} It’s often helpful to visualize the roots on the complex plane. Each fourth root lies at an angle of $\frac{\pi}{12}, \frac{7\pi}{12}, \frac{13\pi}{12}$, and $\frac{19\pi}{12}$ radians from the positive real axis, each spaced $\frac{\pi}{2}$ radians apart (i.e., 90 degrees) scaled by $\sqrt[4]{2}$ (they lie on a circle of radius $\sqrt[4]{2}$).

• What is wrong with $1 = -i² = -i\cdot i = -\sqrt{-1}\cdot\sqrt{-1} = \sqrt{(-1)\cdot (-1)} = \sqrt{1} = 1$?

This is an invalid application of the radical multiplication rule $\sqrt{a}​\cdot \sqrt{b}​=\sqrt{ab}$​. This rule is only valid when at least one of a or b is a non-negative real number. The error is in the illicit application of the rule for multiplying square roots to negative numbers.

In the complex plane, the square root function is multi-valued. For any non-zero complex number z, there are two numbers w such that $w^2 = z$. When dealing with real numbers ($\mathbb{R}$), we define $\sqrt{x}$ to be the principal (positive) root. This convention allows the previous rule to work for positive reals.

Complex-Valued Functions: Definitions, Domain, Range

Complex-valued functions extend the familiar concept of real functions to the complex plane, opening up a rich mathematical landscape with applications in physics, engineering, and pure mathematics.

Definition. Let $D \subseteq \mathbb{C}$ be a set of complex numbers. A complex-valued function f of a complex variable, defined on D, is a rule that assigns to each complex number z belonging to the set D a unique complex number w, $f: D \to \mathbb{C}, z \mapsto w = f(z)$.

Key Concepts

• For a complex function f, the set D of all input values z for which f(z) is defined is called the domain of f, written D = Dom(f). The domain may exclude points where f is not defined or blows up (e.g., f(z) = $\frac{1}{z}$ is undefined at z = 0) or where the expression is multivalued, (e.g., $\sqrt{z}$ or log(z)), unless one restricts to a branch where f becomes single-valued.

  For a complex expression like $w = \sqrt{z}$, each nonzero z has two distinct square roots, so more than one value of w corresponds to the same z. In that sense, $\sqrt{z}$ is not single‐valued on ℂ∖{0}; instead we call it a multiple-valued function. One usually selects a single branch (a single‑valued slice of this set‑valued object).

• The set of all actual outputs {f(z): z ∈ D} is called the range (or image) of the complex function f, also denoted $f(D) = \{f(z) : z \in D\}$. Range can be all of $\mathbb{C}$ (e.g. nonconstant polynomials) or omit values (e.g. exponential never attains 0).
• We often call the elements of D as points. If z = x+ iy ∈ D, then f(z) is called the image of the point z under f.
• We often use $z, \zeta, w$ for complex variables.
• $z = x + iy$ where $x = \Re(z)$ and $y = \Im(z)$
• Every complex function $f: D \rarr \Complex$ can be written in terms of two real-valued functions: $\boxed{f(z) = f(x + iy) = u(x, y) + iv(x, y)}$ where $u: \mathbb{R}^2 \to \mathbb{R}$ is the real part of $f$ and $v: \mathbb{R}^2 \to \mathbb{R}$ is the imaginary part of $f$, e.g., $f(z) = z^3, \forall z \in \mathbb{C}$, $f(x + iy) = x^3 -3xy^2 + i(3x^2y -y^3).~ u(x, y) = x^3 -3xy^2$ is the real part of f and $v(x, y) = 3x^2y -y^3$ is the imaginary part of f.
• A complex function $f: D \subseteq \mathbb{C} \to \mathbb{C}$ can be viewed as: $f: D \subseteq \mathbb{R}^2 \to \mathbb{R}^2, (x, y) \mapsto (u(x, y), v(x, y))$. This connects complex analysis to real multivariable calculus.

Examples

• Constant functions: $f(z) = c, \text{ where } c \in \mathbb{C} \text{ is a constant}$. Domain: any chosen $D \subseteq \mathbb{C}$ (often all of $\mathbb{C}$), e.g., $f(z) = 3 - 2i$ for all $z \in \mathbb{C}$. Range: $\{c\}$ (a single point). Real and imaginary parts: $u(x, y) = \Re(c), v(x, y) = \Im(c)$.
• The Identity Function: $f(z) = z$. Domain: $\mathbb{C}$, Range: $\mathbb{C}$. Real and imaginary parts: $u(x, y) = x, v(x, y) = y$.
• The Conjugate Function: $f(z) = \bar{z}$. Domain: $\mathbb{C}$, Range: $\mathbb{C}$. Real and imaginary parts: $u(x, y) = x, v(x, y) = -y$.

  This function is not holomorphic (complex differentiable). Geometrically, this operation corresponds to a reflection across the real (horizontal) axis.

• The Modulus Function: $f(z) = |z|, f(x + iy) = \sqrt{x^2 + y^2}$ Domain: $\mathbb{C}$, Range: $[0, \infty) \subset \mathbb{R}$. This maps complex numbers to non-negative real numbers.
• Polynomial functions: $f: \Complex \rarr \Complex$ defined by f(z) = $a_nz^n + a_{n-1}z^{n-1} + ··· + a_0$ where n is a non-negative integer (the degree if $a_n \neq 0$), a_n, a_n-1, ··· , a₀ are complex constants, e.g., f(z) = 6z² -(2+4i)z + 12 -$\sqrt{3}i$, a second-degree polynomial (quadratic) since the highest power of $z$ with non-zero coefficient is 2. Domain: $\mathbb{C}$ (entire), Range: $\mathbb{C}$ (for non-constant polynomials). The hightest prior of z carrying a non-zero coefficient is 2, so we say that f is a second degree polynomial.

1. $f(z) = z^2$. $f(x + iy) = (x + iy)^2 = x^2 + 2ixy + (iy)^2 = x^2 - y^2 + 2xyi$. Real and imaginary parts: $u(x, y) = x^2 - y^2, v(x, y) = 2xy$. Concrete evaluation: $f(3 + i) = (3 + i)^2 = 9 + 6i + i^2 = 9 + 6i - 1 = 8 + 6i$. We can always restrict the domain to any particular subset, e.g., the unit disk: $f: B(0, 1) \to \mathbb{C}, f(z) = z^2$ where $B(0, 1) = \{z \in \mathbb{C} : |z| < 1\}$.
2. $f(z) = z^3$. $f(x + iy) = (x + iy)^3 =[\text{Binomial theorem}] = x^3 + 3x^2(iy) + 3x(iy)^2 + (iy)^3= x^3 + 3x^2yi + 3x(-y^2) + (-iy^3) = x^3 - 3xy^2 + i(3x^2y - y^3)$. Real and imaginary parts: $u(x, y) = x^3 - 3xy^2, v(x, y) = 3x^2y - y^3$.

• Rational functions. Let p(z) and q(z) be polynomial functions with complex coefficients, and q(z) is not the zero polynomial $q \not\equiv 0$. Then, $f(z) = \frac{p(z)}{q(z)}$ is called a rational function. The domain of f is the set $\{z \in \mathbb{C} : q(z) \neq 0\}$.
  A powerful property of rational functions is that they can be decomposed into a sum of simpler fractions (partial fractions). Let R(z) = P(z)/Q(z) be a rational function where P and Q are polynomials and deg(P) < deg(Q). Suppose Q(z) factors as $c\prod_{i=1}^m(z-z_i)^{k_i}$ where the $z_i$ are distinct roots of multiplicities $k_i$. Then, R(z) can be written uniquely as $R(z) = \sum_{i=1}^m\sum_{j=1}^{k_i} \frac{A_{ij}}{(z-z_i)^j}$ where the $A_{ij}$ are constants.
  Rational functions are holomorphic (complex differentiable) everywhere in their domain. The singularities occur at the roots of the denominator q(z). However, the nature of the singularity depends on the numerator p(z):

1. Poles If $z_0$​ is a root of q(z) with multiplicity m, but $z_0$ is not a root of p(z), then $z_0$ is a pole of order m, e.g., f(z) = $\frac{z+1}{(z−2)^3}$​ has a pole of order 3 at z=2.
2. Removable Singularities: If $z_0$​ is a root of both p(z) and q(z), it might seem like a singularity, but it is often removable. If $z_0$​ is a root of multiplicity m in q(z) and multiplicity k in p(z), there are two cases:
   If k ≥ m, the singularity is removable (the limit exists), e.g., $f(z) = \frac{z^2-1}{z-1}$. At z = 1, this looks like 0/0, but for $z \ne 1, f(z) = \frac{z^2-1}{z-1} = z + 1$. Thus, z = 1 is indeed a removable singularity.
   If k < m, there is a pole of order m − k.

Some examples:

1. Simple Rational Function. $g(z) = \frac{3z^2 - (2 + i)z}{2z - 3i}$. Domain: $\mathbb{C} \setminus \left\{\frac{3i}{2}\right\}$. Singularity at $z = \frac{3i}{2}$ (pole).
2. Reciprocal Function. $f(z) = \frac{1}{z}$. Domain: $\mathbb{C} \setminus \{0\}$. Singularity at $z = 0$ (simple pole). We can separate the function into real and imaginary components by multiplying numerator and denominator by the complex conjugate: $f(x + iy) = \frac{1}{x + iy} = \frac{x - iy}{(x + iy)(x - iy)} = \frac{x - iy}{x^2 + y^2}$. Real and imaginary parts $u(x, y) = \frac{x}{x^2 + y^2}, v(x, y) = \frac{-y}{x^2 + y^2}$
3. Linear Fractional (Möbius) Transformations. f(z) = $\frac{az+b}{cz+d}$ where $a, b, c, d \in \mathbb{C}$ with $ad - bc \neq 0$ (to ensure non-degeneracy). Domain: $\text{Dom}(f) = \begin{cases} \mathbb{C} \setminus \left\{-\frac{d}{c}\right\}, & c \neq 0 \\ \mathbb{C}, & c = 0, d \neq 0 \end{cases}$
   Example: $g(z) = \frac{3z²-(2+i)z}{2z -3i}$ for each z ∈ ℂ - {$\frac{3i}{2}$}. Properties of Möbius Transformations: (i) Map circles and lines to circles and lines; (ii) Form a group under composition; (iii) They are conformal (angle-preserving).

Algebraic vs Transcendental functions

In complex analysis, functions are broadly classified into algebraic and transcendental based on their relationship to polynomial equations and algebraic operations.

A function f(z) is algebraic if there exists a nonzero polynomial P(z, w) with complex coefficients such that $P(z, f(z)) \equiv 0$ for all z in its domain. Equivalently, algebraic functions are constructed from finitely many algebraic operations (addition, subtraction, multiplication, division, or root extraction) applied to polynomials or rational functions, e.g., polynomials $x^2+3x+5$, rational functions $\frac{2x+1}{x^2-3}$, root functions ($\sqrt{z}=z^{\frac{1}{2}}$ satisfies $w^2-z = 0$; $z^{\frac{1}{n}}$ satisfies $w^n-z =0$) and functions defined implicitly by polynomial equations (e.g., the circle $z^2 + w^2=1$ defines $w = \pm\sqrt{1-z^2}$).

Algebraic functions can be multi-valued (e.g., $\sqrt{z}$ has two branches), but they remain algebraic because each branch satisfies a polynomial equation.

A function is transcendental if it is not algebraic —i.e., it does not satisfy any polynomial relation $P(z, f(z)) \equiv 0$. These functions cannot be expressed using a finite combination of algebraic operations, are inherently multi-valued, and often require infinite processes (infinite power series or integrals, e.g., $e^z = \sum_{n=0}^\infty \frac{z^n}{n!}, log(z) = \int_1^z \frac{dt}{t}$) for their definition. Typical transcendental functions include the complex exponential and logarithmic $e^x, log(z)$, trigonometric (sin(z), cos(z), tan(z)), hyperbolic (sinh(z), cosh(z)), and power functions with non-rational exponents ($z^{\alpha}, \alpha \notin \mathbb{Q}$). They cannot be expressed by a finite combination of algebraic operations.

Single-Valued vs. Multi-Valued Functions

In complex analysis, functions are classified as single-valued or multi-valued based on their input-output behavior:

• A function is single-valued if each input $z$ produces exactly one output $w$, e.g., $z^2$, $e^z$, $\sin z$. In other words, single valued functions assign exactly one output w to each input z.
• A function is multi-valued if some inputs correspond to more than one output. Multi-valued functions produce multiple outputs for some inputs, e.g., $\sqrt{z} = z^{1/2}$ (2 values), $z^{1/n}$ (n values), $\log(z)$ and $z^\alpha$ ($\alpha \notin \mathbb{Q}$) (infinitely many values).

To work with multi-valued functions, we choose a branch cut (curves remove from the complex plane to restrict the function to a single value, e.g., $(-\infty, 0]$ for $\sqrt{z}$ and log(z)) and define principal branches (define specific single-valued versions):

1. For $\sqrt{z}$, the principal branch is $\sqrt{|z|}e^{\frac{iArg(z)}{2}}$ where $Arg(z) \in (-\pi, \pi]$.
2. For log(z), the principal branch is $\textbf{Log(z)} = \textbf{ln|z|} + i(Arg(z))$ where ln denotes the natural logarithm of the magnitude (or modulus) of z and Arg represents the principal argument of z n the range (-π, π].

By choosing a branch cut, multi-valued functions become single-valued and continuous on the cut plane, but they exhibit discontinuity across the branch cut, e.g.:

• $\sqrt{z}$ single-valued and continuous on $\mathbb{C} \setminus (-\infty, 0]$, but discontinuous across the branch cut. Crossing the cut is discontinuous: approaching a point x < 0 from above (upper half-plane, Im(z) > 0, $Arg(z) \to \pi$), $\sqrt{z} \to \sqrt{|z}|e^{\frac{i\pi}{2}} = i\sqrt{|z|}$, but approaching x < 0 from below (lower half-plane, Im(z) < 0, $Arg(z) \to -\pi$), $\sqrt{z} \to \sqrt{|z}|e^{\frac{-i\pi}{2}} = -i\sqrt{|z|}$. These values differ, creating a jump discontinuity across the branch cut.
• Similarly, Log(z) is single-valued and continuous on $\mathbb{C} \setminus (-\infty, 0]$. When crossing $(-\infty, 0]$ from above to below, Arg(z) jumps from $\pi$ to $-\pi$, causing the imaginary part to decrease by $Log(z)_{below} - Log(z)_{above} = (ln|z| -i\pi) -(ln|z| + i\pi) = -2\pi i$

Entire Functions

An entire function is a complex-valued function $f: \mathbb{C} \to \mathbb{C}$ that is holomorphic (complex differentiable) at every point in the complex plane $\mathbb{C}$.

Because complex differentiability is a much stronger condition than real differentiability, entire functions are exceptionally “rigid” and well-behaved (“nice”).

A function f(z) is entire if and only if it satisfies any of the following equivalent conditions:

• Holomorphic everywhere: For all $z_0 \in \mathbb{C}$, the limit $f'(z_0) = \lim_{z \to z_0} \frac{f(z) - f(z_0)}{z - z_0}$ exists and is finite.

• Analytic everywhere: Entire functions can be represented by a convergent power series centered at any point $z_0 \in \mathbb{C}$, with infinite radius of convergence, $f(z) = \sum_{n=0}^\infty a_n(z-z_0)^n$ converges for all $z \in \mathbb{C}$.
• Cauchy-Riemann satisfaction: The real and imaginary parts u(x, y) and v(x, y) satisfy the Cauchy-Riemann equations everywhere and have continuous first partial derivatives.

Examples of Entire Functions

Liouville’s Theorem: If f is an entire function and bounded (i.e., there exists M > 0 such that $|f(z)| \leq M$ for all $z \in \mathbb{C}$), then f is constant.

Proof (via Cauchy’s Integral Formula):

Let f be entire and bounded by M. Let $z_0$ be any point in $\mathbb{C}$. Consider a circle $C_R$ of radius R centered at $z_0$.

By Cauchy’s Integral Formula for the first derivative: $f'(z_0) = \frac{1}{2\pi i} \int_{C_r} \frac{f(\zeta)}{(\zeta - z_0)^2} d\zeta$

Taking the modulus and using the ML-inequality (length of the contour is $2\pi R$, maximum value of integrand is $\frac{M}{R^2}$): $|f'(z_0)| \le \frac{1}{2\pi}\cdot (2\pi R) \cdot \frac{M}{R^2} = \frac{M}{R}$

This inequality holds for any radius R > 0. If we let $R \to \infty$, the right-hand side goes to 0: $|f'(z_0) \le lim_{R \to \infty} \frac{M}{R} = 0 \implies f'(z_0) = 0$.

Since $z_0$ was arbitrarily, $f'(z_0) = 0$ everywhere, therefore f is constant.

Implications and Corollaries

• Fundamental Theorem of Algebra: Every non-constant polynomial P(z) with complex coefficients has at least on root in $\mathbb{C}$.

Proof.

Assume P(z) has no root. Then, g(z) = 1/P(z) is defined everywhere and entire (quotient of entire functions where denominator is never zero).

Since $P(z) \to \infty$ as $|z| \to \infty$, g(z) is bounded. Liouville’s Theorem implies g(z) is constant $\implies$ P(z) is constant, a contradiction.

• Generalized Liouville Theorem. If f is entire and satisfies a growth bound $∣f(z)∣ \le C(1+∣z∣)^k$ for some integer k ≥ 0, then f is a polynomial of degree at most k.

  If the entire function grows no faster than a polynomial, it is a polynomial.

• Picard’s Little Theorem. If f is a non-constant entire function, then its image is either the whole complex plane $\mathbb{C}$ or $\mathbb{C}$ minus a single point, e.g., $e^z$ is entire and never equals 0 (it omits exactly one point); sin(z) and cos(z) actually attain every complex value (they omit nothing).
• Maximum Modulus Principle. If f is entire and non-constant, then ∣f(z)∣ cannot attain a local maximum anywhere in $\mathbb{C}$.
  If ∣f(z)∣ had a global maximum, the function would be bounded, hence constant by Liouville.

Operations on Complex Functions

Complex functions inherit algebraic and analytic properties from real functions.

Arithmetic Operations

If f and g are complex functions, the resulting function’s domain is restricted to where both original functions are defined. Furthermore, if f and g are holomorphic (complex differentiable) on their domains, the resulting function is also holomorphic on the restricted domain (with special care for division).

If f is holomorphic and g is holomorphic and non-zero, f/g is holomorphic. If g has aa zero of order m at $z_0$ and f has a zero of order n at $z_0$:

• If n ≥ m, the singularity is removable.
• If n < m, the function has a pole of order m−n.
• The resulting function is meromorphic (holomorphic except for isolated poles).

Composition

The composition of functions creates a “chain” of domain constraints, $(f \circ g)(z) = f(g(z))$. Domain: $\{z \in \text{Dom}(g) : g(z) \in \text{Dom}(f)\}$

If g is holomorphic at $z_0$ and f is holomorphic at $g(z_0)$ then $f \circ g$ is holomorphic at $z_0$. Derivative (Chain Rule): $(f \circ )'(z) = f'(g(z))\cdot g'(z)$.