If we want to get a Fourier inversion formula with the same measure on both sides ( that is, since we can think about \ R ^ n as its own dual space we can ask for \ widehat { \ mu } to equal \ mu ) then we need to use
32.
There is yet another approach, due to Mogens Flensted-Jensen, which derives the properties of the zonal spherical functions on SL ( 2, "'R "'), including the Plancherel formula, from the corresponding results for SL ( 2, "'C "'), which are simple consequences of the Plancherel formula and Fourier inversion formula for "'R " '.
33.
Furthermore, there is no way to fix which square root of negative one will be meant by the symbol ( it makes no sense to speak of " the positive square root " since only real numbers can be positive, similarly it makes no sense to say " rotation counter-clockwise ", because until is chosen, there is no fixed way to draw the complex plane ), and hence one occasionally sees the Fourier transform written with in the exponent instead of, and vice versa for the inversion formula, a convention that is equally valid as the one chosen in this article, which is the more usual one.
