Even and odd Functions

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Even and odd Functions

in mathematics. The function y = f(x) is said to be even if its value does not change when the sign of the independent variable changes—that is, if f(–x) = f(x). If, however, f(–x) = –f(x), then the function f(x) is said to be odd. For example, y = cos x and y = x² are even functions, and y = sin x, y = x³ are odd functions. The graph of an even function is symmetric with respect to the y-axis, and the graph of an odd function is symmetric with respect to the origin.

References in classic literature ?

Indeed, except in respect of staring about him (a branch of the public service to which the pictorial cherub is much addicted), this domestic cherub discharged as many odd functions as his prototype; with the difference, say, that he performed with a blacking-brush on the family's boots, instead of performing on enormous wind instruments and double-basses, and that he conducted himself with cheerful alacrity to much useful purpose, instead of foreshortening himself in the air with the vaguest intentions.

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Since the orders of magnitude of the amplitudes can be quite different for the even and odd functions, a separation is logical.

The unknown quantities in the diagnostic model are the spatial grid [x.sub.i], the lower and upper spectral boundaries [f.sub.L] and [f.sub.U] of the Fourier transform integral, the even and odd functions [S.sub.E] and [S.sub.O] dependent on frequency f, and the offset.

(a) Cross section of covariances [mathematical expression not reproducible] and [mathematical expression not reproducible] for even and odd function. Each covariance is similar to a sinc function centred at f = f', where a tip is added on top.

Fourier Spectroscopy: A Bayesian Way

Let V be the subspace of odd functions in [L.sup.2.sub.2[pi]] (R).

Since h [member of] [L.sup.2.sub.2[pi]] (R) is an odd function, it can be expressed by the Fourier sine series expansion

Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

Domains of univalence for typically real odd functions. Complex Variables 2003; 48 No.

The problems of n-valence in some classes of analytic functions

It's a thoughtful Proton package, but odd functions let things down a little.

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Symbolic preprocessing is used by the NIntegrate function to simplify piece-wise, even and odd functions. When this preprocessing is not used, Mathematica will generate an automatic "slow convergence (slwcon) message with hints as to the nature of the problem, e.g.

A very significant advance in computational power: Mathematica 6 adds nearly 1,000 new functions, extensive interactive graphics capabilities

y = CV [R.sup.(2)] where CV [R.sup.(2)] is the class of univalent, convex and odd functions in [DELTA] with real coefficients.

On functions convex in the direction of the real axis with real coefficients

T = {f[epsilon] ,A : Im z Im f (z) [greater than or equal to] 0, z [epsilon] [DELTA]}, and its subclass [T.sup.(2)] consisted of odd functions.

the class of typically real odd functions. In fact, if f [member of] [T.sup.(2)] then f [member of] [H.sup.(2)].