Even and odd functions: Difference between revisions

Revision as of 19:00, 8 June 2024

Revision as of 21:15, 8 June 2024

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==Even–odd decomposition==

If a real function has a ___domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the '''even part''' and the '''odd part''' of the function, and. are defined by

<math display =block>f_\text{even}(x) = \frac {f(x)+f(-x)}{2},</math>

and

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This decomposition is unique since, if

:<math>f(x)=g(x)+h(x),</math>

where {{mvar|g}} is even and {{mvar|h}} is odd, then <math>g=f_\text{eeven}</math> and <math>h=f_\text{oodd},</math> since

: <math>\begin{align}

2f_\text{e}(x) &=f(x)+f(-x)= g(x) + g(-x) +h(x) +h(-x) = 2g(x),\\