In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Partial plot of a function f. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. ... Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ...

Even functions

f(x) = x², an example of an even function

Let f(x) be a real-valued function of a real variable. Then f is even if the following equation holds for all real x: Image File history File links Download high resolution version (1162x1162, 15 KB) Un graphique de la fonction x^2 File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Even and odd functions ... Image File history File links Download high resolution version (1162x1162, 15 KB) Un graphique de la fonction x^2 File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Even and odd functions ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

f( − x) = f(x)

The general function to derive the even component of a function is:

Geometrically, an even function is symmetric with respect to the y-axis, meaning that its graph remains unchanged after reflection about the y-axis. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ...

Examples of even functions are | x |, x², x⁴, cos(x), and cosh(x). In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...

Even functions cannot be one-to-one maps. A bijective function. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

Odd functions

f(x) = x, an example of an odd function

Again, let f(x) be a real-valued function of a real variable. Then f is odd if the following equation holds for all real x: Image File history File links Download high resolution version (1162x1162, 16 KB) Une représentation de la fonction x File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Even and odd functions ... Image File history File links Download high resolution version (1162x1162, 16 KB) Une représentation de la fonction x File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): Even and odd functions ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

f( − x) = − f(x)

The general function to derive the odd component of a function is:

Geometrically, an odd function is symmetric with respect to the origin, meaning that its graph remains unchanged after rotation of 180 degrees about the origin. In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ... In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. ... A degree (in full, a degree of arc, arc degree, or arcdegree), usually symbolized °, is a measurement of plane angle, representing 1／360 of a full rotation. ...

Examples of odd functions are x, x³, sin(x), and sinh(x). In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ...

Some facts

Note: A function's being odd or even does not imply differentiability, or even continuity. Properties involving Fourier series, Taylor series, derivatives and so on may only be used when they can be assumed to exist.

Basic properties

• The only function which is both even and odd is the constant function which is identically zero (i.e., f(x) = 0 for all x).
• In general, the sum of an even and odd function is neither even nor odd; e.g. x + x².
• The sum of two even functions is even, and any constant multiple of an even function is even.
• The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
• The product of two even functions is an even function.
• The product of two odd functions is again an even function.
• The product of an even function and an odd function is an odd function.
• The quotient of two even functions is an even function.
• The quotient of two odd functions is an even function.
• The quotient of an even function and an odd function is an odd function.
• The derivative of an even function is odd.
• The derivative of an odd function is even.

In mathematics a constant function is a function whose values do not vary and thus are constant. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

Series

• The Taylor series of an even function includes only even powers.
• The Taylor series of an odd function includes only odd powers.
• The Fourier series of a periodic even function includes only cosine terms.
• The Fourier series of a periodic odd function includes only sine terms.

As the degree of the Taylor series rises, it approaches the correct function. ... Fourier series are a mathematical technique for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude and frequency. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Algebraic structure

• Any linear combination of even functions is even, and the even functions form a vector space over the reals. Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over the reals. In fact, the vector space of all real-valued functions is the direct sum of the subspaces of even and odd functions. In other words, every function can be written uniquely as the sum of an even function and an odd function:

• The even functions form a commutative algebra over the reals. However, the odd functions do not form an algebra over the reals.

In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. ...

Harmonics

In signal processing, harmonic distortion occurs when a sine wave signal is multiplied by a non-linear transfer function. The type of harmonics produced depend on the transfer function^[1]: Signal processing is the processing, amplification and interpretation of signals and deals with the analysis and manipulation of signals. ... The total harmonic distortion, or THD, of a signal is a measurement of the harmonic distortion present and is defined as the ratio of the sum of the powers of all harmonic components to the power of the fundamental. ... In trigonometry, an ideal sine wave is a waveform whose graph is identical to the generalized sine function y = Asin[ω(x − α)] + C, where A is the amplitude, ω is the angular frequency (2π/P where P is the wavelength), α is the phase shift, and C is the... A transfer function is a mathematical representation of the relation between the input and output of a linear time-invariant system. ... In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. ...

• When the transfer function is even, the resulting signal will consist of only even harmonics of the input sine wave;
  • The fundamental is also an odd harmonic, so will not be present.
  • A simple example is a full-wave rectifier.
• When it is odd, the resulting signal will consist of only odd harmonics of the input sine wave;
  • The output signal will be half-wave symmetric.
  • A simple example is clipping in a symmetric push-pull amplifier.
• When it is asymmetric, the resulting signal may contain either even or odd harmonics;
  • A simple example is clipping in an asymmetrical class A amplifier.

The fundamental tone often referred to simply as the fundamental, is the lowest frequency in a harmonic series. ... AC, half-wave and full wave rectified signals A rectifier is an electrical device, comprising one or more semiconductive devices (such as diodes) arranged for converting alternating current to continuous current. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Clipping is one form of distortion that occurs when an amplifier is overdriven, which happens when it attempts to increase voltage or current beyond its limits. ... The term amplifier as used in this article can mean either a circuit (or stage) using a single active device or a complete system such as a packaged audio hi-fi amplifier. ... This topic has already been covered in electronic amplifier. ...

References

1. ^ Ask the Doctors: Tube vs. Solid-State Harmonics

See also

• Taylor series
• Fourier series