I'm to show that sin(x) is orthogonal to sin(x).
From class, we're told that f(x) and g(x) are orthogonal if
I am confused to begin with because when I graph sin(x)sin(x) I see that the area under the curve from -pi to pi is not zero. Can someone help me here please.
The question says "Show that the functions are orthogonal." You should interpret this as meaning that any two distinct functions from that set are orthogonal. A function can only be orthogonal to itself if it is the zero function.
(However, there is a slight inaccuracy in the wording of the question, because it specifies the possibility k=0 for both the sine and the cosine series. The function is the zero function, and so it should probably not have been included.)
Re: signature. It was conjectured by Euler in 1769 that a k'th power cannot be expressed as the sum of fewer than k k'th powers. But Euler, unusually for him, was wrong. The first counterexample was the identity . This was discovered in 1906 (calculated by hand, way before the invention of computers) and published in a volume to commemorate the 400th anniversary of the University of Aberdeen. The identity is more recent (1988). In each of those cases, a k'th power is the sum of k–1 k'th powers. It is still not known whether a k'th power can be the sum of k–2 (or fewer) k'th powers.