Help
Euler defined the function and proved G(n) = n! = 1 middot 2 middot 3 middot middot middot n for all positive integers n. Redo Euler's work by first calculating G(0) = 1 and then use integration by parts to show G(n) = n G(n - 1). That says G(n) = n! by recursion. Using standard methods to reduce to single integrals, evaluate the following double integral in terms of G(m) and G(n): Calculate the Jacobian of the two-variable substitution What is the region S in the ut-plane that maps onto [0, infinity) times [0, infinity) in the xy-plane under the map (u, v) (x, y) = (tu, t(1 - u))? Hint: simplify x + y. Using the substitution in parts (c) and (d) and the change-of-variables theorem, show that Euler used the G(n) integral to invent the definition of n! for non-integers n. Use the result of parts (b) and (e) to find the value of (1/2)!. You may need to evaluate the integral The easiest way to do that is to see that is a very simple curve, whose area is known.