I have two functions and I want to show that they are orthogonal over a given interval. Do I just multiply them together and show that the integral over the interval is zero?
If your definition of inner product is <f, g>= . But there are other ways of defining the inner product in function spaces, usually involving a "weighting function", so the inner product is . For example, the Leguerre polynomials are orthogonal using the inner product defined by . In that case the " is necessary because the interval of definition is infinite. Bessel functions are orthogonal using the inner product defined by . Here, the "x" factor is necessary because the Bessel functions may have a pole of order 1 at 0.
In general, two vectors in an innerproduct space are orthgonal if and only if their inner product is 0. Check to see what inner product you should be using.