Even and odd functions: Difference between revisions

Revision as of 07:27, 11 June 2005

Revision as of 08:43, 21 June 2005

Line 1:

In [[mathematics]], '''even functions''' and '''odd functions''' are [[function (mathematics)|function]]s which satisfy particular [[symmetry]] relations, with respect to taking [[negativeadditive inverse]]s. They are important in many areas of [[mathematical analysis]], especially the theory of [[power series]] and [[Fourier series]].

==Even functions==

Line 7:

:''f''(−''x'') = ''f''(''x'')

Geometrically, an even function is [[symmetry|symmetric]] with respect to the ''y''-axis, meaning that its [[graph of a function|graph]] remains unchanged after [[reflection (mathematics)|reflection]] about the ''y''-axis.

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

Line 15:

==Odd functions==

Again, let ''f''(''x'') be a [[real number|real]] -valued function of a real variable. Then ''f'' is '''odd''' if the following equation holds for all real ''x'':

:''f''(−''x'') = −''f''(''x'')

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

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

Line 48:

===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[[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>

Line 59:

* [[Fourier series]]

[[Category:Set theoryCalculus]]

[[cs:Sudé a liché funkce]]