Odd Function

An odd function is a univariate function satisfying . Geometrically, such functions are symmetric about the origin. Examples of odd functions include , , the sine , hyperbolic sine , tangent , hyperbolic tangent , error function erf , inverse erf , and the Fresnel integrals , and .

For any univariate function whose domain is symmetric about 0, the odd part is defined by

It is odd, and is the even part of . For a univariate polynomial, the odd part is the sum of the monomials containing odd powers of the variable.

An even function times an odd function is odd, and the product of two odd functions is even while the sum or difference of two nonzero functions is odd iff each summand function is odd. The product and quotient of two odd functions is an even function.

If an even function is differentiable, then its derivative is an odd function; what's more, if an odd function is integrable, then its integral over a symmetric interval , , is identically zero. Similarly, if an even function is differentiable, then its derivative is an odd function while the integral of such a function over a symmetric interval is twice the value of its integral over the interval .

Ostensibly, one can define a similar notion for multivariate functions by saying that such a function is odd iff

Even so, such functions are unpredictable and very well may lose many of the desirable geometric properties possessed by univariate functions. Differentiability and integrability properties are similarly unclear.

Since an odd function is zero at the origin, it follows that the Maclaurin series of an odd function contains only odd powers of .