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.
(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.