For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any we have
22.
There is a special case of Lagrange inversion theorem that is used in combinatorics and applies when f ( w ) = w / \ phi ( w ) for some analytic \ phi ( w ) with \ phi ( 0 ) \ ne 0.
23.
The statement that can be reconstructed from is known as the Fourier inversion theorem, and was first introduced in Fourier's " Analytical Theory of Heat ", although what would be considered a proof by modern standards was not given until much later.
24.
One guess I have is to take advantage of the integral definition of the inverse secant, \ arcsec ( x ) = \ int _ 1 ^ x \ frac { dt } { t \ sqrt { t ^ 2-1 } }, expand the square root with a binomial series, antidifferentiate, and then apply the Lagrange inversion theorem, but I do not have experience applying that.
