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Even Part

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The even part p_e(x) of a univariate polynomial p(x) is the sum of the monomials containing even powers of the variable. Equivalently,

p_e(x)=(p(x)+p(-x))/2.

It is an even function, and p(x)-p_e(x) is the odd part of p(x). The same formula defines the even part of any univariate function whose domain is symmetric about 0.

EvenPart

In number theory, the even part Ev(n) of a positive integer n is defined by

Ev(n)=2^(b(n)),

where b(n) is the exponent of the exact power of 2 dividing n. The values for n=1, 2, ..., are 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, ... (OEIS A006519). The even part function can be implemented in the Wolfram Language as 

  EvenPart[0] := 1
  EvenPart[n_Integer] := 2^IntegerExponent[n, 2]

See also

Even Function, Even Power, Greatest Dividing Exponent, Odd Function, Odd Part, Polynomial

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References

Sloane, N. J. A. Sequence A006519/M0162 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Even Part

Cite this as:

Weisstein, Eric W. "Even Part." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EvenPart.html

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