Editing Even and odd functions (section)

==Generalizations==
===Multivariate functions===

'''Even symmetry:'''

A function <math>f: \mathbb{R}^n \to \mathbb{R} </math> is called ''even symmetric'' if:
:<math>f(x_1,x_2,\ldots,x_n)=f(-x_1,-x_2,\ldots,-x_n) \quad \text{for all } x_1,\ldots,x_n \in \mathbb{R}</math>

'''Odd symmetry:'''

A function <math>f: \mathbb{R}^n \to \mathbb{R} </math> is called ''odd symmetric'' if:
:<math>f(x_1,x_2,\ldots,x_n)=-f(-x_1,-x_2,\ldots,-x_n) \quad \text{for all } x_1,\ldots,x_n \in \mathbb{R}</math>

===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. In [[signal processing]], a similar symmetry is sometimes considered, which involves [[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/>

'''Conjugate symmetry:'''

A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''conjugate symmetric'' if
:<math>f(x)=\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>

A complex valued function is conjugate symmetric if and only if its [[real part]] is an even function and its [[imaginary part]] is an odd function.

A typical example of a conjugate symmetric function is the [[cis function]]
:<math>x \to e^{ix}=\cos x + i\sin x</math>

'''Conjugate antisymmetry:'''

A complex-valued function of a real argument <math>f: \mathbb{R} \to \mathbb{C}</math> is called ''conjugate antisymmetric'' if:
:<math>f(x)=-\overline{f(-x)} \quad \text{for all } x \in \mathbb{R}</math>

A complex valued function is conjugate antisymmetric  if and only if its [[real part]] is an odd function and its [[imaginary part]] is an even function.

===Finite length sequences===
The definitions of odd and even symmetry are extended to ''N''-point sequences (i.e. functions of the form <math>f: \left\{0,1,\ldots,N-1\right\} \to \mathbb{R}</math>) as follows:<ref name=ProakisManolakis>{{Citation | last1 =Proakis | first1 =John G. | last2 =Manolakis | first2 =Dimitri G. | title =Digital Signal Processing: Principles, Algorithms and Applications | place =Upper Saddle River, NJ | publisher =Prentice-Hall International | year =1996 | edition =3 | language =en | id =sAcfAQAAIAAJ | isbn =9780133942897 | url-access =registration | url =https://archive.org/details/digitalsignalpro00proa }}</ref>{{rp|p. 411}}

'''Even symmetry:'''

A ''N''-point sequence is called ''conjugate symmetric'' if
:<math>f(n) = f(N-n) \quad \text{for all } n \in \left\{ 1,\ldots,N-1 \right\}.</math>

Such a sequence is often called a '''palindromic sequence'''; see also [[Palindromic polynomial]].

'''Odd symmetry:'''

A ''N''-point sequence is called ''conjugate antisymmetric'' if
:<math>f(n) = -f(N-n) \quad \text{for all } n \in \left\{1,\ldots,N-1\right\}. </math>
Such a sequence is sometimes called an '''anti-palindromic sequence'''; see also [[Palindromic polynomial|Antipalindromic polynomial]].