Even and odd functions: Difference between revisions

Revision as of 08:47, 21 June 2005

Revision as of 08:53, 21 June 2005

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:''f''(−''x'') = −''f''(''x'')

Geometrically, an odd function is symmetric with respect to the [[origin]], meaning that its [[graph of a function|graph]] remains unchanged after [[coordinate rotation|rotation]] of 180 degrees[[degree (angle)|degree]]s about the origin.

The designation '''odd''' is due to the fact that the Taylor series of an odd function includes only odd powers.

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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 [[linear subspace|subspace]]s 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>