This questions will be about the -function (x) = (0,) t^(x1)*e(t) dt. (a) Show that for every positive integer n, (n) = (n 1)!. Thus (x) is a generalized factorial function. (b) We can use the integral definition to define "factorials" for non-integer real numbers. The remaining questions involve showing that (1/2) = . Find a suitable change of coordinates so that (1/2) = 2 (0,) e^(u^2) du. (c) This integral is finite (by comparing to eu) and thus to evaluate it, we can use a slick trick (that you may want to remember). Write I = (0,) e^(u^2) du. Then I^2 = ((0,) e^(u^2) du)^2 = ((0,) e^(u^2) du)((0,) e^(v^2) dv) = e^(u^2 +v^2) du dv Evaluate I^2 using polar coordinates. (d) Conclude the desired result.