INTEGRATION: THE FEYNMAN WAY ANONYMOUS Abstract. In this paper we will learn a common technique not often de-scribed in collegiate calculus courses. After reviewing the necessary theory, we will proceed to work through some typical examples. Throughout this pro-cess, we will see trivial integrals that can be evaluated using basic techniques of integration (such as integration by parts), however we will also encounter inte-grals that would otherwise require more advanced techniques such as contour integration. 1. Introduction Many up-and-coming mathematicians, before every reaching the university level, heard about a certain method for evaluating definite integrals from the following passage in [1]: One thing I never did learn was contour integration. I had learned to do integrals by various methods show in a book that my high school physics teacher Mr. Bader had given me. The book also showed how to differentiate parameters under the integral sign - It’s a certain operation. It turns out that’s not taught very much in the universities; they don’t emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was that, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn’t do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else’s, and they had tried all their tools on it before giving the problem to me. The method Mr. Feynman is referring to often goes by the name of differentiating under the integral sign, differentiation with respect to a parameter, or sometimes even Feynman Integration. However one wishes to name it, the elegance and appeal lies in how this method can be employed to evaluate seemingly complex integrals with nothing more than1 elementary calculus. 1Once one gets past the measure theory required to prove the Theorem 2.1� � � � � � � 2 ANONYMOUS 2. Some Key Theorems The technique of “Feynman Integration” is a simple application of a theorem attributed to Leibniz. In this section we state the theorem in its most basic form, and end by stating a more general version that allows for even weaker hypotheses. In both cases, we address situations where the following equation (which we would love to be true) holds: d ∂ f(x, y)dy = f(x, y)dy. dx Y Y ∂x Before stating these theorems, recall that differentiation is simply a particular ex-ample of a limit insofar as we define df(x) :=: f�(x) := lim f(x + h) − f(x) ,dx h 0 h→with a true definition on the far right. Thus, we see that (2) will hold whenever we may make the following statement, lim f(x, y)dy = lim f (x, y)dy. x a x a→Y Y →Theorem 2.1 (Elementary Calculus Version). Let f : [a, b] ×Y → R be a function, with [a, b] being a closed interval, and Y being a compact subset of Rn . Suppose that both f(x, y) and ∂f (x, y)/∂x are continuous in the variables x and y jointly. Then f(x, y)dy exists as a continuously differentiable function of x on [a, b], with Y derivative � � d dx Y f(x, y)dy = Y ∂ ∂x f(x, y)dy. As mentioned above, the veracity of (2) is completely dependent upon if we can exchange the operations of limiting and integration. If we were to prove the above theorem, our argument would make full use of the compactness of Y , which of course implies uniform continuity. From this fact, we could show that it is justified to switch change the order of limits and integration, thus proving (2). However, in many cases the restriction of compactness can be too severe. Often times we would like Y to be (−∞, a), (a, ∞), (−∞, ∞),etc... In these situations, the following measure theoretic version of the above comes to our rescue: Theorem 2.2 (Measure Theory Version). Let X be an open subset of R, and Ω be a measure space. Suppose f : X × Ω → R satisfies the following conditions: (1) f(x, ω) is a Lebesgue-integrable function of ω for each x ∈ X. (2) For almost all ω ∈ Ω, the derivative ∂f(x, ω)/∂x exists for all x ∈ X. (3) There is an integrable function Θ : Ω R such that ∂f (x, ω)/∂x ≤ Θ(ω) for all x ∈ X. → | | Then for all x ∈ X, d ∂ f(x, ω)dω = f(x, ω)dω. dx Ω Ω ∂x A sketch of the proof of Theorem 2.2 would most likely make some form of a famous result from measure theory, the Dominated Convergence Theorem. This will of course provide us with the justification to switch the order of limit and� INTEGRATION: THE FEYNMAN WAY 3 integration. For the interested reader, we state the theorem whose proof may be found in [5]: Theorem 2.3 (Dominated Convergence Theorem).� Let X be a measure space, and let Φ, f1, f2, . . . be measurable functions such that Φ < ∞ and fn≤ Φ for all X n ∈ N. If fn f a.e., then f is integrable and | | → � � lim fn = f. n→∞ X X Before moving on to some examples, note that among the three criteria in The-orem 2.2, the first two are usually satisfied. Indeed in all of the following examples we need only check criterion 3, i.e. that f is dominated by some integral function. Once we have found the appropriate dominating function, we may safely apply Theorem 2.2 and thus “differentiate under the integral”. 3. Examples In this section we present several examples on the application of the above the-orem(s). We begin with the following basic problem: Example 3.1. Compute the definite integral, � 12x − 1 dx.log x0 In order to apply our theorems, we obviously need to be dealing with an integrand in two variables. In this example, we “generalize” by introducing a parameter b in the exponent of our x term. In particular, we could choose to define the following function: � 1 b I(b) = x − 1 dx.log x As long as b > −1, all conditions of Theorem 2.1 are satisfied and we may differen-tiate under the integral sign: 0 d x ∂ xI�(b) = � 1 b −