Analysis
Power Series and Taylor Expansions
Trigonometric and Hyperbolic Functions
Complex analysis is the study of functions of a complex variable, with a focus on analytic holomorphic functions, contour integration, and their properties. It reveals connections between calculus, topology, and geometry, providing tools and insights for both pure mathematics and applied fields such as physics and engineering. Some results include Cauchy’s integral theorem, residue calculus, and the classification of singularities.
Euler’s formula
Euler’s formula, establishes a profound connection between exponential functions and trigonometric functions. The formula states that for any real number \theta:
e^{i\theta}=\cos (\theta) +i\sin (\theta)
One of the most common and ways to prove Euler’s formula is by utilizing the series expansions of the exponential, cosine, and sine functions.
The Maclaurin series for e^z around z=0 is given by:
e^{z}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}} = 1+{\frac {z}{1!}}+{\frac {z^{2}}{2!}}+{\frac {z^{3}}{3!}}+\cdots
This power series has an infinite radius of convergence, and it defines e^z for all complex numbers z.
To derive Euler’s formula, we substitute z = i\theta into the Taylor series for e^z:
e^{i\theta}=1+i\theta+\frac{i^2\theta^2}{2!}+\frac{i^3\theta^3}{3!}+\frac{i^4\theta^4}{4!}+\frac{i^5\theta^5}{5!}+\frac{i^6\theta^6}{6!}+\frac{i^7\theta^7}{7!}+\frac{i^8\theta^8}{8!}+\cdots
We utilize the powers of the imaginary unit i:
i^ 0 = 1,\; i^1 = i,\; i^2 = -1,\; i^3 = -i,\; i^4 = 1,\; i^5 = i,\; \cdots
and so on, with the pattern repeating every four powers. Substituting these into the series expansion for e^{i\theta} gives:
e^{i\theta}=1+i\theta-\frac{\theta^2}{2!}-\frac{i\theta^3}{3!}+\frac{\theta^4}{4!}+\frac{i\theta^5}{5!}-\frac{\theta^6}{6!}-\frac{i\theta^7}{7!}+\frac{\theta^8}{8!}+\cdots
Now, we can separate the real and imaginary terms:
e^{i\theta} = \left(1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\frac{\theta^6}{6!}+\frac{\theta^8}{8!}-\cdots\right) + i\left(\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\frac{\theta^7}{7!}+\cdots\right)
Recognizing the Maclaurin series for \cos(\theta) and \sin(\theta):
\begin{aligned} \cos(\theta) &= 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \cdots \\ \sin(\theta) &= \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots \end{aligned}
We arrive at Euler’s formula:
e^{i\theta}=\cos (\theta)+i\sin (\theta)
The rearrangement of terms is permissible because each of the series is absolutely convergent.
Euler’s formula has a geometric interpretation in the complex plane. The function e^{i\theta} represents a unit complex number, with its magnitude is 1.
As \theta varies through the real numbers, e^{i\theta} traces out the unit circle in the complex plane. In this context, \theta is the angle (in radians) that the line connecting the origin to the point e^{i\theta} on the unit circle makes with the positive real axis, measured counterclockwise.
Euler’s formula also provides a link to polar coordinates. A complex number z = x + iy can be written in polar form as z = r(\cos\theta + i\sin\theta). Using Euler’s formula, this simplifies to:
z = r e^{i\theta}
Here, r = |z| = \sqrt{x^2 + y^2} is the magnitude (or modulus) of z, and \theta = \text{atan2}(y,x) is the phase of z. The complex conjugate of z, denoted as \overline{z} = x - iy, can be expressed in polar form as:
\overline{z} = r e^{-i\theta}
This polar representation is advantageous for multiplication and exponentiation of complex numbers.
A direct consequence of Euler’s formula is De Moivre’s formula. This theorem states that for any real number \theta and any integer n:
\left( \cos(\theta) + i \sin(\theta) \right)^n =\cos(n\theta) + i \sin(n\theta)
The proof using Euler’s formula is straightforward:
\left( \cos(\theta) + i \sin(\theta) \right)^n = \left(e^{i\theta}\right)^n = e^{in\theta} = \cos(n\theta) + i \sin(n\theta)
De Moivre’s formula shows that raising a complex number to the power of n multiplies its argument (angle) by n and raises its modulus to the power of n (if r \neq 1).
Okay, here is an expanded and more formally presented section on Complex Functions, tailored for an audience with a graduate-level mathematical background:
Complex functions
Let S be a non-empty subset of the complex plane \mathbb{C}. A complex function f with domain S is a mapping f: S \to \mathbb{C} that assigns to each complex number z \in S a complex number w \in \mathbb{C}.
We denote this assignment by w = f(z). The set S is referred to as the domain of definition of f, often denoted \text{dom}(f). The set \{w \in \mathbb{C} \mid w = f(z) \text{ for some } z \in S\} is called the range or image of f, denoted \text{Im}(f) or f(S).
A distinction in complex analysis is made based on the nature of this assignment: if for every z \in S, f(z) yields a single, unique complex number w, then f is termed a single-valued function. This aligns with the standard definition of a function in most mathematical contexts.
However, many important constructions in complex analysis lead to situations where a given z \in S may correspond to a set of multiple distinct complex numbers.
Such a correspondence is termed a multivalued function (or multifunction, multiple-valued mapping). Examples include the complex logarithm and n^{th} root operations.
To rigorously analyze multivalued functions, the concept of branches is introduced. A branch of a multivalued function F(z) is a single-valued function f_k(z) that is continuous (and typically analytic) on some restricted domain S' \subseteq S.
Each branch f_k(z) systematically selects one of the possible values of F(z) for each z \in S'. The process of defining such branches necessitates the introduction of branch cuts – curves or lines in the complex plane that are excluded from the domain S' to ensure the single-valuedness and continuity of the branch.
A principal branch is a conventionally chosen, specific branch of a multivalued function, often denoted with a capitalized initial letter (e.g., \text{Log } z for the principal branch of the complex logarithm). The values yielded by the principal branch are known as principal values.
A complex function f(z) can be expressed by decomposing its value w = u + iv (where u, v \in \mathbb{R}) in terms of the real and imaginary parts of z = x + iy (where x, y \in \mathbb{R}):
f(z) = u(x, y) + iv(x, y)
Here, u(x,y) and v(x,y) are real-valued functions of two real variables, x and y. Properties of f(z) such as continuity, being differentiable or integrable are defined through the properties of u(x,y) and v(x,y).
For example, f(z) is differentiable in the complex sense (analytical) if u and v satisfying the Cauchy-Riemann equations.
Alternatively, if z is expressed in polar coordinates as z = re^{i\theta} (where r > 0 is the modulus and \theta is the argument or angle), the function f(z) can be written as:
f(z) = u(r, \theta) + iv(r, \theta)
In this formulation, u(r, \theta) and v(r, \theta) are real-valued functions of the polar variables r and \theta. This polar representation is often more convenient for functions exhibiting rotational symmetries or for analyzing behaviors around singular points, such as the origin, and for understanding the structure of branches and branch cuts. For example, the argument \theta itself is inherently multivalued, which is directly related to the multivalued nature of functions like the logarithm or roots.
Power functions
A class of single-valued complex functions is the power function, defined for a non-negative integer n as:
f(z) = z^n
where z \in \mathbb{C}. For n=0, we define f(z) = z^0 = 1 for all z \in \mathbb{C} (including z=0).
For n \ge 1, f(z) = z^n signifies multiplying z by itself n times. These functions are entire, meaning they are analytic (complex differentiable) throughout the entire complex plane.
More generally, a polynomial function is a finite linear combination of such power functions:
P(z) = \sum_{k=0}^n a_k z^k = a_n z^n + a_{n-1} z^{n-1} + \dots + a_1 z + a_0
where the coefficients a_k \in \mathbb{C} are complex constants, and n is a non-negative integer called the degree of the polynomial (if a_n \neq 0). Polynomials are also entire functions.
The evaluation of z^n for n \in \mathbb{N}_0 is easy to understand using the polar representation of z. Let z = r e^{i\theta}, where r = |z| is the modulus of z and \theta = \arg(z) is an argument of z. Then, by repeated application of the multiplication rule for complex numbers in polar form we have:
z^n = (r e^{i\theta})^n = r^n (e^{i\theta})^n = r^n e^{in\theta}
Using Euler’s formula, e^{in\theta} = \cos(n\theta) + i\sin(n\theta), we can express z^n in terms of its real and imaginary parts:
f(z) = z^n = r^n \left[ \cos(n\theta) + i \sin(n\theta)\right]
If f(z) = u(r, \theta) + iv(r, \theta), then for the power function z^n:
\begin{aligned} u(r, \theta) & = r^n \cos(n\theta) \\ v(r, \theta) & = r^n \sin(n\theta) \end{aligned}
This representation shows the geometric effect of applying the power function z^n: the modulus r of z is raised to the n^{th} power, and the argument \theta of z is multiplied by n. This implies that a point z in the complex plane is mapped to a point z^n whose distance from the origin is r^n and whose angle with the positive real axis is n\theta.
For integer powers n \ge 0, the function f(z) = z^n is single-valued because although \theta = \arg(z) is multivalued (differing by integer multiples of 2\pi), the term e^{in\theta} remains unique. Specifically, if \theta_0 is the principal argument of z, then any other argument is \theta = \theta_0 + 2k\pi for some integer k:
e^{in\theta} = e^{in(\theta_0 + 2k\pi)} = e^{in\theta_0} e^{in2k\pi} = e^{in\theta_0} (\cos(2nk\pi) + i\sin(2nk\pi)) = e^{in\theta_0} (1 + 0) = e^{in\theta_0}
Therefore, the value of z^n is independent of the choice of k, ensuring z^n is a single-valued function for n \in \mathbb{N}_0.
The concept of power functions extends to negative integer powers z^{-n} = 1/z^n (for z \neq 0) and, more generally, to arbitrary complex powers z^c (where c \in \mathbb{C}), such as z^{1/n} (the n^{th} root function).
However, when the exponent is not an integer, these functions typically become multivalued, requiring the definition of branches, branch points, and branch cuts. For example, z^{1/n} has n distinct values for z \neq 0.
Power series and Taylor expansions
A power series centered at z_0 \in \mathbb{C} is an infinite series of the form:
\sum_{k=0}^\infty a_k (z - z_0)^k = a_0 + a_1(z-z_0) + a_2(z-z_0)^2 + \dots
where z is a complex variable and a_k \in \mathbb{C} are complex coefficients.
If a complex function f(z) is analytic in an open disk D(z_0, r) = \{z \in \mathbb{C} : |z-z_0| < r\} centered at z_0 with radius r > 0, then f(z) can be uniquely represented by a convergent power series within that disk.
This series is its Taylor series expansion around z_0:
f(z) = \sum_{k=0}^\infty \frac{f^{(k)}(z_0)}{k!} (z - z_0)^k
Here, f^{(k)}(z_0) denotes the k^{th} complex derivative of f evaluated at z_0. The coefficients are given by:
a_k = \frac{f^{(k)}(z_0)}{k!}
A function f is said to be analytic on an open set D \subseteq \mathbb{C} if for every z_0 \in D, f(z) can be expressed as a power series centered at z_0 that converges to f(z) in some neighborhood of z_0 (i.e., in an open disk D(z_0, \epsilon) for some \epsilon > 0).
A function is complex analytic in an open set D if and only if it is holomorphic in D, meaning it is complex differentiable at every point in D. Consequently, the terms “analytic” and “holomorphic” are often used interchangeably.
This equivalence implies that if a function is complex differentiable once in an open set, it is infinitely differentiable in that set, and its Taylor series converges to the function.
Every power series has a radius of convergence, denoted by r, which is a non-negative real number or \infty. This radius defines a disk of convergence D(z_0, r) = \{z \in \mathbb{C} : |z-z_0| < r\} such that:
- The series converges absolutely for every z such that |z-z_0| < r.
- The series converges uniformly on any compact subset of D(z_0, r).
- The series diverges for every z such that |z-z_0| > r.
The behavior of the series on the boundary circle |z-z_0| = r (when 0 < r < \infty) is generally more complex and requires separate investigation.
The radius of convergence r is the radius of the largest open disk centered at z_0 within which the series converges. For a function f(z) analytic at z_0, its Taylor series centered at z_0 converges to f(z) within the largest open disk centered at z_0 where f(z) is analytic.
This means the radius of convergence of the Taylor series is the distance from z_0 to the nearest singularity of f(z) (a point where f(z) fails to be analytic).
If f(z) has no singularities in the finite complex plane, it is an entire function (e.g., e^z, \sin z, \cos z, polynomials), and the radius of convergence of its Taylor series around any z_0 is r = \infty.
The radius of convergence r can be determined using the Cauchy-Hadamard theorem, which involves the limit superior:
r = \frac{1}{\limsup_{k\to\infty} \sqrt[k]{|a_k|}}
If the limit superior is 0, then r = \infty. If the limit superior is \infty, then r = 0 (meaning the series converges only at z=z_0).
The root test states that the series \sum a_k (z-z_0)^k converges if \limsup_{k\to\infty} \sqrt[k]{|a_k(z-z_0)^k|} < 1, which is equivalent to |z-z_0| \limsup_{k\to\infty} \sqrt[k]{|a_k|} < 1. This directly yields the formula for r.
Alternatively, if the limit exists, the ratio test often provides a more straightforward way to compute r:
r = \lim_{k\to\infty} \left|\frac{a_k}{a_{k+1}}\right|
This formula is derived from the condition for convergence in the ratio test:
\lim_{k\to\infty} \left|\frac{a_{k+1}(z-z_0)^{k+1}}{a_k(z-z_0)^k}\right| < 1 This simplifies to:
|z-z_0| \lim_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| < 1
The series converges if:
|z-z_0| < \frac{1}{\lim_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right|} = \lim_{k\to\infty} \left|\frac{a_k}{a_{k+1}}\right|
assuming the limit exists.
Consider the Maclaurin series (Taylor series centered at z_0 = 0) for f(z) = e^z:
e^z = \sum_{k=0}^\infty \frac{z^k}{k!}
Here, a_k = \frac{1}{k!}. Using the ratio test for the radius of convergence:
r = \lim_{k\to\infty} \left|\frac{a_k}{a_{k+1}}\right| = \lim_{k\to\infty} \left|\frac{1/k!}{1/(k+1)!}\right| = \lim_{k\to\infty} \left|\frac{(k+1)!}{k!}\right| = \lim_{k\to\infty} |k+1| = \infty
This confirms that the exponential function is an entire function, and its Taylor series converges for all z \in \mathbb{C}.
Trigonometric and hyperbolic functions
The familiar trigonometric and hyperbolic functions from real calculus can be extended to the complex domain. Their definitions and properties reveal interconnections, primarily through the exponential function.
The complex sine and cosine functions are defined via their Maclaurin series, which converge for all z \in \mathbb{C}:
\begin{aligned} & \sin(z) = \sum_{k=0}^\infty \frac{(-1)^k z^{2k+1}}{(2k+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots \\ & \cos(z) = \sum_{k=0}^\infty \frac{(-1)^k z^{2k}}{(2k)!} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \cdots \end{aligned}
These definitions lead directly to extensions of Euler’s formula. By manipulating Euler’s formula, e^{iz} = \cos(z) + i\sin(z), and e^{-iz} = \cos(-z) + i\sin(-z) = \cos(z) - i\sin(z), we can express the complex sine and cosine functions in terms of the complex exponential:
\begin{aligned} \sin(z) & = \frac{e^{iz} - e^{-iz}}{2i} \\ \cos(z) & = \frac{e^{iz} + e^{-iz}}{2} \end{aligned}
Other trigonometric functions like \tan(z) = \sin(z)/\cos(z), \cot(z) = \cos(z)/\sin(z), \sec(z) = 1/\cos(z), and \csc(z) = 1/\sin(z) are defined in terms of sine and cosine, analogous to their real counterparts. These functions will have singularities where their denominators are zero.
Similarly, the complex hyperbolic sine, cosine, and tangent functions are defined using the complex exponential function:
\begin{aligned} & \sinh(z) = \frac{e^{z} - e^{-z}}{2} \\ & \cosh(z) = \frac{e^{z} + e^{-z}}{2} \\ & \tanh(z) = \frac{\sinh(z)}{\cosh(z)} = \frac{e^{z} - e^{-z}}{e^{z} + e^{-z}} = \frac{e^{2z} - 1}{e^{2z} + 1} \end{aligned}
Other hyperbolic functions like \coth(z), \operatorname{sech}(z), and \operatorname{csch}(z) are defined analogously.
A set of identities connects these complex trigonometric and hyperbolic functions, highlighting their underlying unity:
\begin{aligned} & i \sin(z) = \sinh(iz) \\ & \cos(z) = \cosh(iz) \\ & \sin(iz) = i \sinh(z) \\ & \cos(iz) = \cosh(z) \end{aligned}
Furthermore, the exponential function itself can be expressed in terms of hyperbolic functions:
e^{z} = \cosh(z) + \sinh(z)
And using the relationships above, this also connects back to trigonometric functions with imaginary arguments:
e^{z} = \cos(iz) - i\sin(iz)
The Maclaurin series for \sin(z) and \cos(z) are provided above. One could apply the ratio test or the Cauchy-Hadamard formula to their coefficients to determine their radii of convergence, similar to the procedure shown for the exponential function.
For \sin(z), the coefficients are:
a_{2k+1} = \frac{(-1)^k}{(2k+1)!}
and a_{2k}=0. For \cos(z):
a_{2k} = \frac{(-1)^k}{(2k)!}
and a_{2k+1}=0.
However, a more straightforward approach leverages their definitions in terms of the complex exponential function. Since e^z is an entire function (analytic everywhere, with an infinite radius of convergence), and e^{iz} is simply a composition of e^w with w=iz (which is also entire), it follows that e^{iz} and e^{-iz} are entire functions.
As \sin(z) and \cos(z) are linear combinations of e^{iz} and e^{-iz} and finite linear combinations of entire functions are themselves entire, both \sin(z) and \cos(z) are entire functions.
Consequently, their Taylor series expansions (including their Maclaurin series) converge for all z \in \mathbb{C}, meaning their radius of convergence is R = \infty.
The same reasoning applies to the complex hyperbolic functions \sinh(z) and \cosh(z). Since they are defined as linear combinations of e^z and e^{-z}, which are entire, \sinh(z) and \cosh(z) are also entire functions with an infinite radius of convergence for their Taylor series.
Therefore, it is not necessary to recalculate the radius of convergence for each of these trigonometric and hyperbolic functions using tests on their series coefficients; their entire nature can be deduced directly from their relationship to the exponential function.
Complex logarithm
The complex logarithm, denoted \operatorname{Log } (z), is defined as the inverse of the complex exponential function. That is, for a non-zero complex number z, w = \operatorname{Log } (z) if and only if z = e^w. Understanding the complex logarithm requires careful consideration of its multivalued nature, which arises directly from the periodicity of the complex exponential function.
Let z = x + iy, which can be expressed in polar form as z = r e^{i\theta}, where r = |z| = \sqrt{x^2+y^2} is the modulus of z and \theta = \operatorname{arg}(z) is an argument (or phase) of z. Let w = u + iv be the complex logarithm of z. Then, by definition:
z = e^w = e^{u+iv} = e^u e^{iv}
Comparing the polar form of z with e^u e^{iv}:
r e^{i\theta} = e^u e^{iv}
For these two complex numbers to be equal, their moduli must be equal, and their arguments must be equal (or differ by an integer multiple of 2\pi):
Equating moduli: r = e^u. Since r = |z| is a positive real number, u is uniquely determined as the real natural logarithm of r:
u = \ln(r) = \ln|z| Note that z cannot be zero, as |z|=r would be zero, and \ln(0) is undefined. Thus, \operatorname{Log } 0 is undefined.
Equating arguments: e^{i\theta} = e^{iv}. This implies that v must be an argument of z.
Since the complex exponential e^{i\phi} is periodic with period 2\pi i (e^{i\phi} = e^{i(\phi + 2n\pi)} for any integer n), if \theta_p = \operatorname{Arg}(z) is the principal value of the argument of z (typically chosen such that -\pi < \theta_p \le \pi), then any argument \theta of z can be written as \theta = \theta_p + 2n\pi for some integer n \in \mathbb{Z}.
Therefore,
v = \theta = \operatorname{arg}(z) = \operatorname{Arg}(z) + 2n\pi, \quad n \in \mathbb{Z}
Combining these results, the complex logarithm \operatorname{Log } (z) is a multivalued function given by:
\operatorname{Log } (z) = \ln|z| + i(\operatorname{Arg}(z) + 2n\pi), \quad n \in \mathbb{Z}
For a single non-zero complex number z, there are infinitely many possible values for \operatorname{Log } (z), each differing by an integer multiple of 2\pi i. These different values correspond to different “sheets” in the Riemann surface of the logarithm.
The image illustrates how multiple values in the w-plane, differing by 2\pi i in their imaginary part, map to the same z value via the exponential function, implying the multivalued nature of its inverse, the logarithm.
To work with the logarithm as a single-valued function, as required for many applications like defining analytic functions, we must select a specific branch. A branch of \operatorname{Log } (z) is a single-valued, continuous (and typically analytic) function obtained by restricting the argument \theta to a specific interval of length 2\pi.
The most common choice is the principal value (or principal branch) of the logarithm, denoted by capital \operatorname{Log } (z). It is defined by choosing n=0 and taking the principal value of the argument, \operatorname{Arg}(z), which is conventionally restricted to the interval (-\pi, \pi]. Thus:
\operatorname{Log } (z) = \ln|z| + i\operatorname{Arg}(z) where \operatorname{Arg}(z) \in (-\pi, \pi].
The restriction of the argument to an interval like (-\pi, \pi] makes the function \operatorname{Log } (z) single-valued.
However, this introduces a discontinuity. As z approaches a point on the negative real axis from the upper half-plane, \operatorname{Arg}(z) \to \pi. As z approaches the same point from the lower half-plane, \operatorname{Arg}(z) \to -\pi.
This jump in the imaginary part of \operatorname{Log } (z) by 2\pi i means that \operatorname{Log } (z) is not continuous along the negative real axis (including z=0, where it’s undefined anyway).
This line of discontinuity is called a branch cut. For the principal value \operatorname{Log } (z), the branch cut is conventionally chosen to be the non-positive real axis: \{z \in \mathbb{C} \mid \operatorname{Re}(z) \le 0, \operatorname{Im}(z) = 0\}.
By removing the branch cut from the domain, i.e., considering \mathbb{C} \setminus (-\infty, 0], the principal branch \operatorname{Log } (z) becomes an analytic function on this restricted domain.
The mapping for the principal branch can be described as:
\operatorname{Log} : \mathbb{C} \setminus (-\infty, 0] \to \{w = u+iv \in \mathbb{C} \mid u \in \mathbb{R}, v \in (-\pi, \pi)\}
If the endpoint \pi is included in the range of \operatorname{Arg}(z), then the codomain for v is (-\pi, \pi]. The strict inequality is often used when defining the domain of analyticity.
The image depicts the imaginary part of the principal value of the complex logarithm, \operatorname{Im}(\operatorname{Log } (z)) = \operatorname{Arg}(z). The discontinuity along the negative real axis, where the value jumps from \pi to -\pi, is clearly visible, illustrating the branch cut.
Other branches of the logarithm can be defined by choosing different intervals for the argument (e.g., (0, 2\pi]) or, equivalently, by selecting different values of n in the general formula. Each such choice will result in a different branch cut.
The point z=0 is a branch point for \operatorname{Log } (z), as is z=\infty; encircling these points leads from one branch to another. Each branch of \operatorname{Log } (z) is analytic on its domain of definition (i.e., \mathbb{C} excluding the branch cut and the origin).
Complex rational powers
The concept of raising a complex number z to a power a can be extended from integer exponents to rational exponents a \in \mathbb{Q}, and more generally to any complex exponent.
When a is rational, z^a is typically a multivalued function, with its definition rooted in the complex exponential and logarithm functions.
Let a \in \mathbb{Q}. We can write a = m/n, where m \in \mathbb{Z}, n \in \mathbb{N}^+ (positive integers), and without loss of generality, we can assume that m and n are coprime (i.e., their greatest common divisor, \operatorname{gcd}(m,n), is 1). The complex power z^a is defined as:
z^a = z^{m/n} = e^{\frac{m}{n} \operatorname{Log}(z)}
where \operatorname{Log}(z) is the multivalued complex logarithm. Recall that \operatorname{Log}(z) = \ln|z| + i(\operatorname{Arg}(z) + 2\pi k) for k \in \mathbb{Z}, where r = |z| is the modulus of z and \theta_p = \operatorname{Arg}(z) is the principal value of the argument of z (typically -\pi < \theta_p \le \pi).
Substituting the expression for \operatorname{Log}(z):
\begin{aligned} z^{m/n} & = e^{\frac{m}{n} (\ln r + i(\theta_p + 2\pi k))} \\ & = e^{\frac{m}{n} \ln r + i\frac{m}{n}\theta_p + i\frac{2\pi mk}{n}} \\ & = e^{\frac{m}{n} \ln r} \cdot e^{i\frac{m}{n}\theta_p} \cdot e^{i\frac{2\pi mk}{n}} \\ & = r^{m/n} e^{i\frac{m}{n}\theta_p} e^{i\frac{2\pi mk}{n}} \end{aligned} Here, r^{m/n} denotes the principal real root (\sqrt[n]{r})^m. The first two factors, r^{m/n} and e^{i\frac{m}{n}\theta_p}, are single-valued for a given z (once \theta_p is fixed). The multivalued nature of z^{m/n} arises from the term e^{i\frac{2\pi mk}{n}} as k ranges over the integers.
We need to determine how many distinct values this term produces. The values e^{i\phi_1} and e^{i\phi_2} are identical if and only if \phi_1 - \phi_2 is an integer multiple of 2\pi. Thus, we are looking for distinct values of \frac{2\pi mk}{n} modulo 2\pi, which is equivalent to finding distinct values of \frac{mk}{n} modulo 1.
Since m and n are coprime, as k takes values 0, 1, 2, \dots, n-1, the products mk will take on n distinct values modulo n. Consequently, the term e^{i\frac{2\pi mk}{n}} will yield exactly n distinct values.
For k \ge n or k < 0, these values will repeat. For example, if k=n, e^{i\frac{2\pi mn}{n}} = e^{i2\pi m} = (e^{i2\pi})^m = 1^m = 1, which is the same value obtained for k=0.
Therefore, for a rational exponent a = m/n (in lowest terms), z^a has exactly n distinct values, given by:
(z^{m/n})_k = r^{m/n} \exp\left[i\left(\frac{m}{n}(\theta_p + 2\pi k)\right)\right] \quad \operatorname{for } k = 0, 1, \dots, n-1 or equivalently,
(z^{m/n})_k = r^{m/n} e^{i\frac{m}{n}\theta_p} e^{i\frac{2\pi mk}{n}} \quad \operatorname{for } k = 0, 1, \dots, n-1
The principal value of z^a is usually defined by taking k=0, which corresponds to using the principal value of the logarithm \operatorname{Log } z:
\operatorname{PV}(z^a) = r^a e^{ia\operatorname{Arg}(z)} Each of the n values can be thought of as belonging to a different branch of the function f(z)=z^a. Defining a single-valued analytic branch requires introducing a branch cut (typically along the non-positive real axis, inherited from \operatorname{Log } z).
A common special case is finding the n^{th} roots of unity, i.e., 1^{1/n}. Here z=1, so r=|1|=1 and \theta_p = \operatorname{Arg}(1)=0. The exponent is a=1/n, so m=1.
The n distinct n^{th} roots of unity are:
(1^{\frac{1}{n}})_k = 1^{\frac{1}{n}} e^{i\frac{1}{n}(0)} e^{i\frac{2\pi k}{n}} = e^{i\frac{2\pi k}{n}} \quad \operatorname{for } k = 0, 1, \dots, n-1 These values are:
\omega_k = \cos\left(\frac{2\pi k}{n}\right) + i\sin\left(\frac{2\pi k}{n}\right)
and they lie on the unit circle in the complex plane, forming the vertices of a regular polygon with n sides and n vertex.
For example, consider \sqrt[3]{1}:
\begin{aligned} & {e^{i\frac{1}{3}(2\pi k)}}_{|k=0} = e^{i0} = 1\\ & {e^{i\frac{1}{3}(2\pi k)}}_{|k=1} = e^{i\frac{2\pi}{3}} = \left(\frac{2}{3}\pi\right)+ i\sin\left(\frac{2}{3}\pi\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\\ & {e^{i\frac{1}{3}(2\pi k)}}_{|k=2} = e^{i\frac{4\pi}{3}} = \cos\left(\frac{4}{3}\pi\right) + i\sin\left(\frac{4}{3}\pi\right) = -\frac{1}{2} - i\frac{\sqrt{3}}{2}\\ & {e^{i\frac{1}{3}(2\pi k)}}_{|k=3} e^{i2\pi} = e^{i0} = 1\\ & \cdots \end{aligned}
If the exponent a is an irrational real number (e.g., a=\pi) or a general complex number not purely rational, then z^a = e^{a \operatorname{Log}(z)} will generally have infinitely many distinct values.
This is because the term e^{ia(2\pi k)} will not repeat with a finite period if a is not rational, or the modulus |e^{a \operatorname{Log}(z)}| itself may vary with k if a has a non-zero imaginary part.