Even and odd functions: Difference between revisions

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In [[mathematics]], '''even functions''' and '''odd functions''' are functions 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==

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

An '''even function''' is a function which satisfies the condition:

:<math>''f''(-&minus;''x'') = ''f''(''x'')</math>

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

The denominationdesignation '''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''<mathsup>x^2</mathsup>, ''x''<mathsup>x^4</mathsup>, <math>[[trigonometric function|cos]](''x'')</math>, <math>and [[hyperbolic function|cosh]](''x'')</math>.

* The double [[derivative]] of any even function is an even function.

* The [[multiplication|product]] of 2 even functions is an even function.

* The [[addition|sum]] of 2 even functions is an even function.

==Odd functions==

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

An '''odd function''' is a function which satisfies the condition:

:''f''(&minus;''x'') = &minus;''f''(''x'')

Geometrically, an odd function is symmetric with respect to the [[origin]].

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

Examples of odd functions are ''x'', ''x''³, [[trigonometric function|sin]](''x''), and [[hyperbolic function|sinh]](''x'').

==Some facts==

===Basic properties===

:<math>f(-x) = -f(x)</math>

Therefore* anThe oddonly function which is symmetric''both'' witheven respectand toodd is the originconstant function which is identically zero.

* In general, the 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 double [[derivative]] of anyan even function is odd, and the derivative of an evenodd function is even.

===Algebraic Structure===

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

* 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 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.

Examples of odd functions are <math>x</math>, <math>x^3</math>, <math>sin(x)</math>, <math>sinh(x)</math>.

* The even functions form a [[algebra over a field|commutative algebra]] over the reals. However, the odd functions do ''not'' form an algebra over the reals.

==See also==

* [[Taylor_seriesTaylor series]]

* [[Fourier series]]