Odd part
From Calculus
Contents
Definition
Suppose is a function whose domain is a subset of the reals that is symmetric about 0, i.e., for every in the domain of , is also in the domain of . Then, the odd part of , sometimes denoted or is defined as a function with the same domain, and with the definition:
Equivalently, it is the only possible choice of odd function in a decomposition of of the form:
with both having the same domain as , and with an even function and an odd function. The other part, , is the even part of .
Facts
Effect on even and odd functions
- The odd part of an even function is the zero function on the same domain.
- The odd part of an odd function is the same function.
Properties preserved on taking the odd part
Effect of operators on odd part
Operator | Effect on odd part (short version) | Effect on odd part (in symbols) | Proof | |
---|---|---|---|---|
pointwise sum | the odd part of a sum of functions is the sum of the odd parts of each function | Failed to parse (syntax error): (f + g)_o = f_o + g[[odd part operator is linear]] |} ==Particular cases== {| class="sortable" border="1" ! Function !! Domain !! Odd part |- | [[polynomial function]] || all reals || sum of all the monomials of odd degree in the polynomial |- | [[exponential function]] <math>e^x | all reals | hyperbolic sine function |