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Even and odd functions

Contents
Basic properties
    Uniqueness
    Addition and subtraction
    Multiplication and division
    Composition
    Inverse function
Even–odd decomposition
Further algebraic properties
Analytic properties
    Basic analytic properties
    Series
Harmonics
    Complex-valued functions
References

The sine function and all of its Taylor polynomials are odd functions. The cosine function and all of its Taylor polynomials are even functions.

In mathematics, an even function is a real function such that f(−x)=−f(x) for every y-axis, and odd functions are those whose graph is self-symmetric with respect to the origin.

If the domain of a real function is self-symmetric with respect to the origin, then the function can be uniquely decomposed as the sum of an even function and an odd function.

f is even if, for every f(−x)=f(x) or equivalently x↦x2,
  • cos⁡,
  • hyperbolic cosine f(x)=x3 is an example of an odd function.

    A real function −x is also in its domain and x=0 is in the domain of an odd function x↦sgn⁡(x),

  • The identity function n,
  • sin⁡,
  • hyperbolic sine f(x)=x3+1 is neither even nor odd.

    Basic properties

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    Uniqueness

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    • If a function is both even and odd, it is equal to 0 everywhere it is defined.
    • If a function is odd, the absolute value of that function is an even function.

    Addition and subtraction

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    • The sum of two even functions is even.
    • The sum of two odd functions is odd.
    • The difference between two odd functions is odd.
    • The difference between two even functions is even.
    • The sum of an even and odd function is not even or odd, unless one of the functions is equal to zero over the given domain.

    Multiplication and division

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    • The product and quotient of two even functions is an even function.
      • This implies that the product of any number of even functions is also even.
      • This implies that the reciprocal of an even function is also even.
    • The product and quotient of two odd functions is an even function.
    • The product and both quotients of an even function and an odd function is an odd function.
      • This implies that the reciprocal of an odd function is odd.

    Composition

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    • The composition of two even functions is even.
    • The composition of two odd functions is odd.
    • The composition of an even function and an odd function is even.
    • The composition of any function with an even function is even (but not vice versa).

    Inverse function

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    • If an odd function is invertible, then its inverse is also odd.

    Even–odd decomposition

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    If a real function has a domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the even part (or the even component) and the odd part (or the odd component) of the function, and are defined by feven is even, f(x)=g(x)+h(x), where g=feven and ex=cosh⁡(x)⏟feven(x)+sinh⁡(x)⏟fodd(x).

    Fourier's sine and cosine transforms also perform even–odd decomposition by representing a function's odd part with sine waves (an odd function) and the function's even part with cosine waves (an even function).

    Further algebraic properties

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    • 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 functions is the direct sum of the subspaces of even and odd functions. This is a more abstract way of expressing the property in the preceding section.
      • The space of functions can be considered a graded algebra over the real numbers by this property, as well as some of those above.
    • The even functions form a commutative algebra over the reals. However, the odd functions do not form an algebra over the reals, as they are not closed under multiplication.

    Analytic properties

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    A function's being odd or even does not imply differentiability, or continuity. For example, the Dirichlet function is even, but is nowhere continuous.

    In the following, properties involving derivatives, Fourier series, Taylor series are considered, and these concepts are thus supposed to be defined for the considered functions.

    Basic analytic properties

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    • The derivative of an even function is odd.
    • The derivative of an odd function is even.
    • If an odd function is integrable over a bounded symmetric interval [−A,A], the integral over that interval is twice the integral from 0 to A; that is
      ∞ converges.

    • Series

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    Harmonics

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    In signal processing, harmonic distortion occurs when a sine wave signal is sent through a memory-less nonlinear system, that is, a system whose output at time t only depends on the input at time t and does not depend on the input at any previous times. Such a system is described by a response function 0f component represents the DC offset, due to the one-sided nature of even-symmetric transfer functions.

  • When it is odd, the resulting signal will consist of only odd harmonics of the input sine wave; f:ℝn→ℝ is called even symmetric if: f(x1,x2,…,xn)=−f(−x1,−x2,…,−xn)for all x1,…,xn∈ℝ

    Complex-valued functions

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    The definitions for even and odd symmetry for complex-valued functions of a real argument are similar to the real case. In signal processing, a similar symmetry is sometimes considered, which involves complex conjugation.

    Conjugate symmetry:

    A complex-valued function of a real argument x→eix=cos⁡x+isin⁡x

    Conjugate antisymmetry:

    A complex-valued function of a real argument f:{0,1,…,N−1}→ℝ) as follows:

    Even symmetry:

    A N-point sequence is called conjugate symmetric if

    1. 1 2 W., Weisstein, Eric. "Even Function". mathworld.wolfram.com.{{cite web}}: CS1 maint: multiple names: authors list (link)
    2. ↑ Proakis, John G.; Manolakis, Dimitri G. (1996), Digital Signal Processing: Principles, Algorithms and Applications (3 ed.), Upper Saddle River, NJ: Prentice-Hall International, ISBN 9780133942897, sAcfAQAAIAAJ

    • References

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