Even and odd functions: Difference between revisions

Revision as of 03:27, 14 April 2004

Revision as of 03:37, 14 April 2004

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In [[mathematics]], '''even functions''' and '''odd functions''' are functions[[function (mathematics)|function]]s which satisfy particularly nice [[symmetry]] relations. They are important in many areas of [[mathematical analysis]], especially the theory of [[power series]] and [[Fourier series]].

==Even functions==

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Geometrically, an even function is [[symmetry|symmetric]] with respect to the ''y''-axis.

The designation '''even''' is due to the fact that the [[Taylor series]] of an even function includes only even powers and the [[Fourier series]] of an even function includes only [[trigonometric function|cosine]] terms.

Examples of even functions are ''x''², ''x''⁴, [[trigonometric function|cos]](''x''), and [[hyperbolic function|cosh]](''x'').

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===Basic properties===

* The only function which is ''both'' even and odd is the [[constant function]] which is identically zero.

* In general, the [[addition|sum]] of an even and odd function is neither even nor odd; e.g. ''x'' + ''x''².

* The [[addition|sum]] of 2 even functions is even, and any constant multiple of an even function is even. Also, The [[addition|sum]] of 2 odd functions is odd, and any constant multiple of an odd function is odd.

* The [[multiplication|product]] of 2 even functions is an even function; however, the product of 2 odd functions is also an '''even''' function.

* The [[derivative]] of an even function is odd, and the derivative of an odd function is even.

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===Algebraic Structure===

* Any [[linear combination]] of even functions is even, and the even functions form a [[vector space]] over the [[real number|real]]s. 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 spaces 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:

:<math>f(x)=\frac{f(x)+f(-x)}{2}\,+\,\frac{f(x)-f(-x)}{2}</math>