Even and odd functions: Difference between revisions

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===Complex-valued functions===

The definitions for even and odd symmetry for [[Complex number|complex-valued]] functions of a real argument are similar to the real case but involve [[complex conjugation]].<ref name=Oppenheim>

{{Cite book |last1=Oppenheim |first1=Alan V. |author-link=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |year=1999 |publisher=Prentice Hall |___location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |page=55

}}</ref><ref name=ProakisManolakis/>{{rp|p.291}}

'''Even symmetry:'''

A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''evenconjugate symmetric'' if:

:<math>f(x)=\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>

'''Odd symmetry:'''

A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''oddconjugate symmetricantisymmetric'' if:

:<math>f(x)=-\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>