Even and odd functions: Difference between revisions

Revision as of 17:03, 15 April 2004

Revision as of 19:11, 15 April 2004

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==Some facts==

===Basic properties===

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* 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 sum of 2 even functions is even, and any constant multiple of an even function is even. Also,
* The 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.

* The product of 2 odd functions is again an even function.

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

* The derivative of an odd function is even.

===Series===

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* 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 ana periodic even function includes only [[trigonometric function|cosine]] terms.

* The [[Fourier series]] of ana periodic odd function includes only [[trigonometric function|sine]] terms.

===Algebraic Structure===