290 MATHEMATICS MAGAZINE Four Ways to Evaluate a Poisson Integral HONGWEI CHEN Christopher Newport University Newport News, VA 23606 hchen@pcs.cnu.edu In general, it is difficult to decide whether or not a given function can be integrated in elementary ways. In light of this, it is quite surprising that the value of the Poisson integral /7 I(x)= ln( - 2xcos + x2) dO can be determined precisely. Even more surprising is that we can do so for every value of the parameter x. Using four different methods, we will show that 0, if Ixl < 1; )27rlnlxl, if xl > 1. Our integral is one of several known as the Poisson Integral; all are related in some way to Poisson's integral formula, which recovers an analytic function on the disk from its boundary values, a relationship we mention below. However, none of our methods involves complex analysis at all. The first one uses Riemann sums and relies on a trigonometric identity. The second method is based on a functional equation and involves a sequence of integral substitutions. The third method uses parametric differ- entiation and the half-angle substitution. We finish with an approach based on infinite series. It is interesting to see how wide a range of mathematical topics are exploited. These evaluations are suitable for an advanced calculus class and provide a very nice application of Riemann sums, functional equations, parametric differentiation, and in- finite series. We begin with three elementary observations: 1. I (0) = 0. 2. I(-x)= I(x). 3. I(x) = 2r In I x I + I(l/x), (x g: 0). The readercan probably supply the proofs for these, but we will demonstratethe third. If x :7 0, we have 7r x 2 1 I(x) = ln x2 1- - cos 0 + do = r) = 2lnx I(x). =/ y lnx2dO + I(l/x) = 2rrln Ix I + I(l/x). In view of this third observation, our main formula follows easily once we show that I (x) = 0 for I x I < 1. This will be the goal of the next four sections. I. Using Riemann sums Since 1 -2xcos 0 + 2 > (1 - Ix 1)2, for I x < 1, 290 MATHEMATICS MAGAZINE Mathematical Association of America is collaborating with JSTOR to digitize, preserve, and extend access to Mathematics Magazine www.jstor.org ®