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Even and odd functions

In mathematics, even functions are those that satisfy the condition f(-x) = f(x) for all x in their domain, meaning their graphs are symmetric about the y-axis. Odd functions, in contrast, satisfy f(-x) = -f(x), resulting in graphs symmetric about the origin. These classifications apply to functions from the real numbers to the real numbers and are fundamental in analyzing symmetry in calculus and beyond. Graphically, an even function mirrors itself across the y-axis, so if the point (x, y) lies on the graph, then (-x, y) does as well. For odd functions, the about the implies that if (x, y) is on the graph, then (-x, -y) is also present, and such functions must pass through the with f(0) = 0. Algebraically, one tests for evenness or oddness by substituting -x into the function and comparing the result to f(x) or -f(x). Common examples include polynomials where even powers like x^2 or x^4 yield even functions, while odd powers such as x or x^3 produce functions. follow suit: \cos(ax) is even, and \sin(ax) is . A function that is both even and odd must be the zero function everywhere it is defined. Any can be uniquely decomposed into its even part, \frac{f(x) + f(-x)}{2}, and odd part, \frac{f(x) - f(-x)}{2}. Key properties include rules for products: the product of two even functions or two odd functions is even, while the product of an even and an function is odd. In integration, the integral of an function over a symmetric [-L, L] is zero, whereas for an even function, it equals twice the integral from 0 to L. These symmetries extend to applications in , where even functions expand using only cosines and functions using only sines, simplifying series representations.

Definitions and Examples

Even Functions

An even function is a univariate real-valued function f that satisfies the condition f(-x) = f(x) for every x in its domain. This definition captures the inherent symmetry of the function with respect to input negation. Geometrically, the graph of an even function exhibits mirror symmetry across the y-axis, meaning that the point (x, f(x)) on the graph has a corresponding point (-x, f(x)) that is its reflection over the y-axis. This property allows the function's behavior to the right of the y-axis to precisely mirror its behavior to the left. Common examples illustrate this symmetry clearly. The f(x) = x^2 is even, since f(-x) = (-x)^2 = x^2 = f(x), and its forms a parabola opening upward, symmetric about the y-axis. The f(x) = \cos x is also even, as \cos(-x) = \cos x, resulting in a periodic wave that repeats symmetrically on both sides of the y-axis. Additionally, the f(x) = |x| qualifies as even because |-x| = |x|, producing a V-shaped that is perfectly mirrored across the y-axis. Even functions are typically defined on domains symmetric about zero, such as intervals of the form (-a, a) where a > 0, to ensure the symmetry condition applies consistently across the entire . This symmetry contrasts with that of odd functions, which exhibit antisymmetry about the origin.

Odd Functions

An odd function is a function f defined on a domain symmetric about such that f(-x) = -f(x) for all x in the domain. This property implies that the function values at x and -x are negatives of each other, reflecting an antisymmetric behavior with respect to the origin. Geometrically, the graph of an odd function exhibits point symmetry about the origin, meaning that rotating the graph 180 degrees around the origin maps it onto itself. This rotational symmetry distinguishes odd functions from even functions, which are symmetric about the y-axis. For the definition to hold, the domain must be symmetric about zero, so that if x is in the domain, then -x is also in the domain. If zero is in the domain of an odd function, then f(0) = -f(0), which implies f(0) = 0. Common examples include the f(x) = x, which passes through the and satisfies the condition linearly; the sine function f(x) = \sin x, a periodic odd function with antisymmetry in each cycle; and the f(x) = x^3, which maintains odd symmetry while curving away from the . Constant functions f(x) = c where c \neq 0 satisfy f(-x) = f(x), making them even rather than , while the zero function f(x) = [0](/page/0) is both even and .

Algebraic Properties

Basic Operations

The basic operations on even and odd functions preserve or alter in predictable ways, depending on the parities of the functions involved. For and , the or of two even functions is even. To see this, let f and g be even functions, so f(-x) = f(x) and g(-x) = g(x). Then, (f + g)(-x) = f(-x) + g(-x) = f(x) + g(x) = (f + g)(x), and similarly, (f - g)(-x) = f(-x) - g(-x) = f(x) - g(x) = (f - g)(x). The or of two functions is : if f and g are , then f(-x) = -f(x) and g(-x) = -g(x), so (f + g)(-x) = -f(x) - g(x) = -(f(x) + g(x)) = -(f + g)(x), and (f - g)(-x) = -f(x) - (-g(x)) = -f(x) + g(x) = -(f(x) - g(x)) = -(f - g)(x). However, the or of an even and an is neither even nor in general, unless one is the zero . For multiplication, the product of two even functions or two odd functions is even, while the product of an even function and an odd function is odd. Let f and g be even: then (f \cdot g)(-x) = f(-x) g(-x) = f(x) g(x) = (f \cdot g)(x). $$ If both are odd, (f \cdot g)(-x) = (-f(x)) (-g(x)) = f(x) g(x) = (f \cdot g)(x). $$ For one even and one odd, say f even and g odd, (f \cdot g)(-x) = f(-x) g(-x) = f(x) (-g(x)) = - (f(x) g(x)) = - (f \cdot g)(x). Division follows analogous rules where defined, but requires the denominator to be non-zero and the to be symmetric about the to preserve the function's classification. The of two even functions or two odd functions is even: for even f and g with g(x) \neq 0, \left( \frac{f}{g} \right)(-x) = \frac{f(-x)}{g(-x)} = \frac{f(x)}{g(x)} = \left( \frac{f}{g} \right)(x), and similarly for two odd functions. The of an even function by an odd function (or vice versa) is odd.

Composition and Inverse

The composition of even and odd functions exhibits specific properties that can be determined by examining the behavior under the x \to -x. Let f and g be functions defined on appropriate domains such that the f \circ g is well-defined. The of f \circ g is analyzed as follows. If both f and g are even, then (f \circ g)(-x) = f(g(-x)) = f(g(x)) = (f \circ g)(x), so f \circ g is even. For example, the composition of f(x) = x^2 (even) and g(x) = \cos x (even) yields f(g(x)) = \cos^2 x, which is even. If f is even and g is odd, then (f \circ g)(-x) = f(g(-x)) = f(-g(x)) = f(g(x)) = (f \circ g)(x) since f is even, so f \circ g is even. An example is f(x) = |x| (even) composed with g(x) = x^3 (odd), giving |x^3| = |x|^3, which is even. In general, any even function composed with any function (even or odd) results in an even function. If f is odd and g is even, then (f \circ g)(-x) = f(g(-x)) = f(g(x)) = (f \circ g)(x), so f \circ g is even. For instance, f(x) = x^3 (odd) composed with g(x) = x^2 (even) gives (x^2)^3 = x^6, which is even. If both f and g are odd, then (f \circ g)(-x) = f(g(-x)) = f(-g(x)) = -f(g(x)) = -(f \circ g)(x), so f \circ g is odd. An example is f(x) = \sin x (odd) composed with g(x) = x (odd), yielding \sin x, which is odd. Even functions are generally not invertible over domains symmetric about the , such as \mathbb{R}, because they fail to be : for x \neq 0, f(x) = f(-x) implies multiple inputs map to the same output. In contrast, functions that are bijective possess that are also . To see this, suppose y = f(x) where f is and invertible. Then -y = f(-x), so applying f^{-1} gives f^{-1}(-y) = -x = -f^{-1}(y). For example, the inverse of f(x) = x^3 () is f^{-1}(y) = y^{1/3}, which satisfies f^{-1}(-y) = (-y)^{1/3} = - (y^{1/3}) = -f^{-1}(y) and is thus .

Even-Odd Decomposition

Uniqueness of Decomposition

In , any function f defined on a domain symmetric about the origin—such as [-a, a] or \mathbb{R}—admits a unique into an even function e and an odd function o such that f = e + o. This establishes that every such function can be expressed as the sum of components with distinct properties, where even functions satisfy e(-x) = e(x) and odd functions satisfy o(-x) = -o(x). To outline the proof of existence, the even component is given by e(x) = \frac{f(x) + f(-x)}{2}, and the odd component by o(x) = \frac{f(x) - f(-x)}{2}. Direct substitution confirms that e(x) + o(x) = f(x). Furthermore, e(-x) = e(x) verifies the evenness of e, while o(-x) = -o(x) confirms the oddness of o. The uniqueness follows from the fact that the only function that is both even and odd is the zero function. Suppose f = e_1 + o_1 = e_2 + o_2, where e_1, e_2 are even and o_1, o_2 are odd. Then e_1 - e_2 = o_2 - o_1. The left side is even, while the right side is odd, implying that both sides equal zero (as their sum would otherwise contradict parity). Thus, e_1 = e_2 and o_1 = o_2. This unique decomposition enables parity-based analysis for arbitrary functions, even without inherent symmetry, which simplifies computations in integration, differentiation, and series expansions by isolating even and odd behaviors.

Construction Methods

The even and odd parts of a function f defined on a domain symmetric about the origin can be explicitly constructed using the averaging formulas: e(x) = \frac{f(x) + f(-x)}{2}, o(x) = \frac{f(x) - f(-x)}{2}. These expressions yield the even component e(x) and the odd component o(x), satisfying f(x) = e(x) + o(x). As an illustrative example, consider f(x) = x + x^2. Then f(-x) = -x + x^2, so e(x) = \frac{(x + x^2) + (-x + x^2)}{2} = x^2, o(x) = \frac{(x + x^2) - (-x + x^2)}{2} = x. The even part x^2 is symmetric about the y-axis, while the odd part x exhibits antisymmetry about the origin. This decomposition separates the polynomial into its naturally even and odd terms. The formulas ensure the constructed parts satisfy the required . For the even part, e(-x) = \frac{f(-x) + f(x)}{2} = e(x), demonstrating y-axis . For the odd part, o(-x) = \frac{f(-x) - f(x)}{2} = -\frac{f(x) - f(-x)}{2} = -o(x), confirming point about the . These verifications follow directly from the definitions and the of . If the of f is not about the , such as [0, \infty), the formulas require extension of f to a symmetric domain via reflection to define values at negative arguments. An even reflection sets f(-x) = f(x) for x > 0, mirroring the graph over the y-axis, while an odd reflection sets f(-x) = -f(x), reflecting through the origin. The choice of extension influences the resulting parts but enables application of the construction on the full symmetric domain. Graphically, this decomposition isolates symmetric and antisymmetric behaviors. The even part e(x) appears as the y-axis-symmetric portion of f(x), formed by averaging the original with its y-axis , emphasizing mirrored features across the vertical axis. The odd part o(x) captures the origin-symmetric component, obtained by averaging the original with its 180-degree about the , highlighting opposing values at x and -x. This method provides the unique even-odd decomposition, as justified by the .

Analytic Properties

Differentiation and Integration

If a function f is even and differentiable at a point x, then its f' is at that point. To see this, consider the definition of the : f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}. For f'(-x), substitute into the limit: f'(-x) = \lim_{h \to 0} \frac{f(-x + h) - f(-x)}{h}. Since f is even, f(-x + h) = f(x - h) and f(-x) = f(x), so f'(-x) = \lim_{h \to 0} \frac{f(x - h) - f(x)}{h} = \lim_{k \to 0} \frac{f(x + k) - f(x)}{-k} \cdot (-1) = -f'(x), where k = -h. Thus, f'(-x) = -f'(x), confirming that f' is . Conversely, if f is odd and differentiable at x, then f' is even. The proof follows similarly: substituting the odd condition f(-x + h) = -f(x - h) and f(-x) = -f(x) into the derivative limit for f'(-x) yields f'(-x) = \lim_{h \to 0} \frac{-f(x - h) + f(x)}{h} = -\lim_{h \to 0} \frac{f(x - h) - f(x)}{h} = f'(x), after the substitution k = -h. Examples include f(x) = \sin x, which is odd with even derivative f'(x) = \cos x. These properties hold under the assumption of differentiability, typically requiring continuity of f at the point. For integration, the parity of a function affects definite integrals over symmetric intervals [-a, a] where a > 0. If f is even and continuous on [-a, a], then \int_{-a}^{a} f(x) \, dx = 2 \int_{0}^{a} f(x) \, dx. This arises from the symmetry: the areas from -a to 0 and 0 to a are equal due to f(-x) = f(x). For instance, with f(x) = x^2, the integral from -1 to 1 is $2 \int_0^1 x^2 \, dx = \frac{2}{3}. If f is odd and continuous on [-a, a], then \int_{-a}^{a} f(x) \, dx = 0. The positive and negative areas cancel due to f(-x) = -f(x). An example is f(x) = x, where \int_{-1}^{1} x \, dx = 0. More generally, for any integrable f = e + o where e is the even part and o the odd part, the integral decomposes as \int_{-a}^{a} f(x) \, dx = 2 \int_0^a e(x) \, dx, with the odd part contributing zero. These results extend to Riemann or Lebesgue integrability, assuming the integral exists.

Series Representations

If an even function has a expansion centered at the , it contains only even powers of the variable, while if an odd function has such an expansion, it contains only odd powers. For instance, the power series for the cosine function, which is even, is given by \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}, valid for all real x. Similarly, the sine function, which is odd, has the expansion \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!} x^{2n+1}, also valid for all real x. The Taylor series expansion of a function at zero, known as the Maclaurin series, directly reflects the parity of the function through its derivatives evaluated at the origin. For an even function f, all odd-order derivatives vanish at zero (f^{(k)}(0) = 0 for odd k), resulting in a series with only even powers. Conversely, for an odd function f, all even-order derivatives (except possibly the zeroth, where f(0) = 0) vanish at zero (f^{(k)}(0) = 0 for even k > 0), yielding only odd powers in the series. This property arises because the derivative of an even function is odd, and vice versa, leading to the alternation and vanishing conditions at the symmetric point zero. In Fourier series representations of periodic functions over symmetric intervals like [-\pi, \pi], even functions expand solely in terms of cosine functions, which form an even basis. The series takes the form f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx), where the coefficients a_n are computed via integrals that exploit the of cosines. Odd functions, by contrast, expand only in sine terms, an odd basis, yielding f(x) = \sum_{n=1}^{\infty} b_n \sin(nx), with sine coefficients b_n from corresponding orthogonal integrals. Any sufficiently periodic function can be decomposed into its even and odd parts, and its naturally separates these components: the cosine terms capture the even part, while the sine terms represent the odd part. This decomposition leverages the of the trigonometric basis, allowing the even and odd contributions to be isolated independently in the expansion.

Applications

Harmonic Functions

In the context of physics, even and odd functions play a fundamental role in describing vibrations. Symmetric vibrations, which exhibit mirror about the position, are modeled using even functions such as cosines, corresponding to even harmonics that maintain the same sign on both sides of the axis. In contrast, antisymmetric vibrations, which invert across the axis, are represented by odd functions like , aligning with odd harmonics that produce opposite displacements on either side. This distinction arises naturally from the properties required to satisfy conditions in equations for oscillatory systems. Decomposition of periodic functions into even and odd components in simplifies the study of in wave phenomena. By separating a into its even part (yielding only cosine terms) and odd part (yielding only sine terms), the resulting series representation highlights how symmetric and antisymmetric behaviors contribute to the overall signal structure. This approach is particularly useful in , where the even-odd split reduces and reveals underlying symmetries in the . For instance, the even component corresponds to basis functions that are cosine-like, aiding in the identification of balanced harmonic contributions without cross-terms from opposite symmetries. A representative example appears in standing waves on a taut string fixed at both ends, where modes are classified by their symmetry about the . Odd modes are symmetric, with mirror-image displacements across the center and antinodes at the , such as the first harmonic where the entire string moves in phase without a central . Even modes, conversely, are antisymmetric, with displacements inverting across the center, as seen in the second harmonic with a at the and equal but opposite lobes on either side. These symmetries directly tie to even and function properties in the mode shapes, facilitating solutions to the wave equation via . The conceptual framework of even and odd functions in harmonics traces back to 18th- and 19th-century developments in solving the . Jean le Rond d'Alembert's 1747 formulation of the one-dimensional for vibrating strings incorporated initial conditions that preserved even or odd , enabling general solutions via traveling waves. Leonhard Euler extended this work in the 1750s by applying series expansions—precursors to —to arbitrary initial displacements, implicitly leveraging even-odd symmetries to match boundary conditions and decompose vibrations. These contributions laid the groundwork for using even and odd functions as basis elements in harmonic solutions to wave problems. In modern applications, such as audio synthesis, the distinction between even and odd harmonics influences tonal qualities. Even harmonics, derived from even function components, generate "warm" tones by adding consonant overtones that blend smoothly with the fundamental, evoking richness in analog-style sounds. Odd harmonics, stemming from odd functions, produce "bright" or edgy timbres through dissonant higher frequencies that enhance clarity and presence, commonly used to sharpen synthesized waveforms. This selective emphasis on even or odd components allows engineers to tailor harmonic content for desired auditory effects in music production.

Signal Processing

In signal processing, even signals exhibit symmetry about the time origin, satisfying x(t) = x(-t) for all t, while odd signals demonstrate antisymmetry, satisfying x(t) = -x(-t). These properties are leveraged in and , where separating signals into even and odd components enables efficient processing, such as in symmetric responses that reduce computational demands in linear systems. Any arbitrary signal can be uniquely decomposed into even and odd parts using x_e(t) = \frac{x(t) + x(-t)}{2} and x_o(t) = \frac{x(t) - x(-t)}{2}, allowing manipulation that preserves overall while simplifying operations like time reversal and scaling. In the (DFT) and its fast implementation (FFT), the spectrum of an even signal is real-valued and even, effectively reducing to a , whereas an signal's spectrum is purely imaginary and , corresponding to a discrete sine transform. This from even- decomposition facilitates signal compression by enabling sparse representations and reduced data storage, as seen in transform-based coding schemes where only real or imaginary components need processing. For instance, in electrocardiogram (ECG) analysis, even-odd splitting within lifting-based wavelet transforms decomposes noisy signals into subsampled even and odd subsets, enabling targeted denoising that suppresses artifacts like baseline wander while retaining diagnostic features such as QRS complexes. In image processing, even-symmetric components in phase congruency models, derived from pairs, enhance by identifying points of maximum phase alignment, improving robustness to noise and illumination variations in applications like imagery. Even-odd properties also streamline convolution, a core operation in filtering: the convolution of two even signals or two odd signals yields an even result, while an even signal convolved with an odd signal produces an odd result, allowing parity to guide efficient kernel designs and output predictions without full recomputation. Software tools support these techniques in practice; MATLAB's Signal Processing Toolbox includes utilities like fft for spectral analysis and custom scripts for even-odd extraction via averaging over time-reversed copies, while Python's SciPy library offers scipy.signal functions for decomposition and symmetry-based filtering in DSP workflows.

Generalizations

Multivariate Real Functions

In the context of functions from \mathbb{R}^n to \mathbb{R}, a function f is defined as even if f(-\mathbf{x}) = f(\mathbf{x}) for all \mathbf{x} \in \mathbb{R}^n, and odd if f(-\mathbf{x}) = -f(\mathbf{x}) for all \mathbf{x} \in \mathbb{R}^n. This extends the univariate notion of parity symmetry to vector arguments, where -\mathbf{x} denotes component-wise negation. Functions of multiple variables often exhibit partial symmetries, being even or odd with respect to specific variables while independent of parity in others. For instance, the function f(x,y) = x^2 y is even in x since f(-x,y) = (-x)^2 y = x^2 y = f(x,y), but odd in y because f(x,-y) = x^2 (-y) = -x^2 y = -f(x,y). Such mixed parities arise naturally when analyzing dependencies on individual coordinates. In physics, these concepts appear under parity transformations, where position-dependent quantities like V(\mathbf{r}) are even functions, satisfying V(-\mathbf{r}) = V(\mathbf{r}) due to the scalar nature of the potential and its dependence on the magnitude |\mathbf{r}|. Conversely, momentum \mathbf{p} transforms as an odd vector, with \mathbf{p}(-\mathbf{r}) = -\mathbf{p}(\mathbf{r}), reflecting its pseudovector behavior under spatial inversion.) These parities ensure conservation laws and simplify symmetry analyses in classical and . Properties of even and odd multivariate functions extend component-wise from the univariate case: the sum of even functions is even, the sum of odd functions is odd, and the product of two even or two odd functions is even. However, mixed parities complicate compositions and products, as the overall depends on the specific symmetries involved—for example, the product of an even and an odd function is odd. For integration over domains symmetric about the , such as balls or hypercubes centered at zero, the of an odd function vanishes: \int_D f(\mathbf{x}) \, d\mathbf{x} = 0 if D = -D. This follows from pairing points \mathbf{x} and -\mathbf{x}, where contributions cancel due to the antisymmetry f(-\mathbf{x}) = -f(\mathbf{x}). Even functions, by contrast, yield twice the over the positive subdomain.

Complex-Valued Functions

In the theory of , even and odd functions are generalized from their real-variable counterparts to functions f: \Omega \to \mathbb{C}, where \Omega \subseteq \mathbb{C} is an open symmetric about the . A function f is defined as even if it satisfies f(-z) = f(z) for all z \in \Omega, and if f(-z) = -f(z) for all z \in \Omega. These definitions preserve the symmetry properties under inversion through the origin, but analyticity imposes additional constraints: if f is holomorphic (complex differentiable) in \Omega, its or expansion around 0 contains only even powers for even functions and only powers for functions. For instance, the e^z is neither even nor odd, as e^{-z} = e^z only holds trivially at specific points and e^{-z} \neq -e^z in general. In contrast, the cosine function \cos z = \frac{e^{iz} + e^{-iz}}{2} is even, satisfying \cos(-z) = \cos z, while the sine function \sin z = \frac{e^{iz} - e^{-iz}}{2i} is , satisfying \sin(-z) = -\sin z; both are entire (holomorphic everywhere in \mathbb{C}). A related but distinct symmetry involves complex conjugation, leading to Hermitian and anti-Hermitian functions. A function f is Hermitian even if f(\bar{z}) = \overline{f(z)} for all z \in \Omega, meaning it maps conjugates to conjugates, analogous to real-valued functions on the real axis. Conversely, an anti-Hermitian odd function satisfies f(\bar{z}) = -\overline{f(z)}, reflecting an antisymmetry under conjugation. These properties are particularly relevant for functions that are analytic in domains symmetric with respect to the real axis, as they ensure compatibility with the Cauchy-Riemann equations, which express holomorphicity via \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} for f(z) = u(x,y) + iv(x,y). For even or odd analytic functions, the parity condition further symmetrizes the real part u and imaginary part v: in an even holomorphic function, both u(-x,-y) = u(x,y) and v(-x,-y) = v(x,y); for odd, u(-x,-y) = -u(x,y) and v(-x,-y) = -v(x,y). This interplay ensures that the Cauchy-Riemann equations hold alongside the parity symmetry without additional singularities. Such functions find significant applications in , where the \hat{P} acts on wavefunctions \psi(z) by \hat{P} \psi(z) = \psi(-z), classifying states as even (eigenvalue +1) or odd (eigenvalue -1) under spatial inversion. In systems with inversion , such as the or , the yields solutions with definite , simplifying the analysis of selection rules in and processes; for example, even-parity wavefunctions couple only to even , preserving the . This framework extends to complex representations of quantum states, where holomorphic wavefunctions in inherit even or odd properties to model coherent states or -protected topological phases.

Discrete Sequences

In and , the concepts of even and odd functions extend to indexed by integers, adapting the properties to bidirectional or finite indexing. A \{a_n\}_{n=-\infty}^{\infty} is defined as even if a_{-n} = a_n for all integers n, exhibiting about n=0; it is odd if a_{-n} = -a_n for all n, showing antisymmetry about n=0 with a_0 = 0. For finite sequences of length N, even and odd symmetries are defined relative to the center of the sequence, often in the context of periodic extensions used in transforms like the (DFT). A real-valued \{x(n)\}_{n=0}^{N-1} is symmetric (even) if x(n) = x(N - n) for n = 1, \dots, N-1, requiring only \lfloor N/2 + 1 \rfloor independent values to specify it fully. It is antisymmetric () if x(0) = 0 and x(N - k) = -x(k) for k = 1, \dots, N-1, determined by \lceil N/2 - 1 \rceil independent values. In the DFT, the N-point transform of an even yields purely real coefficients that are symmetric (X(k) = X(N - k)), while an produces purely imaginary coefficients that are antisymmetric (X(k) = -X(N - k)). Representative examples include cosine sequences, which are even (\cos(\omega_k n)), and sine sequences, which are odd (\sin(\omega_k n)), leading to real and imaginary DFT components, respectively; this symmetry simplifies computations in frequency-domain analysis. Any discrete sequence can be uniquely decomposed into its even and odd parts as a_n = \frac{a_n + a_{-n}}{2} + \frac{a_n - a_{-n}}{2}, where the first term is even and the second is odd; for real-valued sequences, these parts are orthogonal. Convolution of discrete sequences preserves in a manner analogous to continuous functions: the convolution of two even sequences is even, of two odd sequences is even, and of an even and an odd sequence is . In the Z-transform domain, an even sequence \{x\} with x = x[-n] satisfies X(z) = X(z^{-1}), implying pole-zero symmetry where poles and zeros occur in reciprocal pairs, and for real-valued even sequences, they appear in complex-conjugate reciprocal quartets. These properties facilitate efficient analysis in and related fields.