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 , the error function , 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 and the sum or difference of two odd functions is odd. The converse need not hold. The quotient of two odd functions, where the denominator is nonzero, is an even function.

If an odd function is differentiable, then its derivative is an even function. Similarly, the derivative of a differentiable even function is an odd function. If an odd function is integrable over a symmetric interval , where , then its integral over is identically zero. If an even function is integrable over , then its integral over is twice its integral over the interval . For , the two one-sided improper integrals must each be finite.

The same notion extends to multivariate functions by saying that such a function is odd iff

On a domain symmetric about the origin, each partial derivative of a differentiable odd multivariate function is even under simultaneous sign reversal. The integral of an integrable odd function over such a domain in is zero.

If an odd function has a Maclaurin series, that series contains only odd powers of .