You may have already encountered the factorial function in your study of mathematics. It is defined for all nonnegative integers according to the following scheme:

1. First, we'll start with an easy example. Consider the function defined according to the following expression:

   f(p) = ó õ ¥ 1  1 xp  dx.

   Be careful! The variable is p.
   1. Find the domain of f. You'll have to look at three cases separately: p < 1, p = 1, and p > 1.
   2. Sketch the graph of f.
   3. Find the point c such that f(c) = c. This point is called a fixed point. Do you understand why?
   4. Show that

      f¢(p) = - ó õ ¥ 1  lnx xp  dx.

      (Hint: Since the integration is with respect to x, you may differentiate with respect to p through the integral.)
2. Evaluate ò₀^¥ x^p-1 e^-x  dx for p = 1,2,3,4,5,6. (Hint: If you do this by hand, look for a pattern.)
3. Make a conjecture regarding the value of ò₀^¥ x^p-1e^-x  dx for any integer p > 0.
4. The function you began studying in Problem 2 is called the gamma function:

   G(p) = ó õ ¥ 0  xp-1 e-x  dx,        p > 0.

   Use integration-by-parts to show that G(p+1) = p G(p).
5. It can be shown that G(¹/₂) = (p)^1/2. Use this fact and the recurrence relation above to compute G(1.5), G(2.5), and G(3.5).
6. The gamma function arises in many applications, but it is not always apparent. Express the following integrals in terms of the gamma function.
   1. ò₀^¥ x² e^-x2  dx,        Hint: Let u = x²
   2. ò₀¹ x² ( ln¹/_x )³  dx,        Hint: Let x = e^-u
7. Write a paragraph describing the history of the gamma function.