In this case, however, since odd functions are being used, a change of sign is also included.

The product of two odd functions is an even function.

First note that y is an odd function of x.

However, so there cannot be a continuous odd function whose range is .

BUT has several equivalent statements in terms of odd functions.

We consider states that are represented by even or odd functions with respect to behavior under space inversion.

The property means that the error function is an odd function.

If is a continuous odd function, then there exists an such that: .

It is automatic, for instance, if K is an odd function.

The Fourier series of a periodic odd function includes only sine terms.

The Maclaurin series of an odd function includes only odd powers.

The integrand is an odd function of , so it vanishes.

Then we can construct another odd function by:

The difference between two odd functions is odd.

That's an odd function for an augmentation.

Let be a continuous odd function.

Thus, is an odd function.

See Even and odd functions.

However, the odd functions do not form an algebra over the reals, as they are not closed under multiplication.

All similarity-invariant solutions are odd functions.

Fourier series; expansions for even and odd functions; half-range expansions.

Because the labeling is an odd function on the boundary, the boundary must have both positive and negative labels:

Both cube and cube root are odd functions:

Again we assume a coordinate transformation as where x are even functions and θ are odd functions.

With these definitions, is an even function and its derivative with respect to z, ℘', is an odd function.