Integration Using Schwinger Parametrization

I recently read the terrific article Integration Using Schwinger Parametrization, by David M. Bradley, Albert Natian, and Sean M. Stewart in the American Mathematical Monthly. I won’t reproduce the entire article here, but I’ll hit a couple of early highlights.

The basic premise of the article is that a complicated integral can become tractable by changing it into an apparently more complicated double integral. The idea stems from the gamma integral

$\Gamma(p) = \displaystyle \int_0^\infty t^{p-1} e^{-t} \, dt$ ,

where $\Gamma(p) = (p-1)!$ if $p$ is a positive integer. If we perform the substitution $t = \phi u$ in the above integral, where $\phi$ is a quantity independent of $t$ , we obtain

$\Gamma(p) = \displaystyle \int_0^\infty (\phi u)^{p-1} e^{-\phi t} \phi \, du = \displaystyle \int_0^\infty \phi^p u^{p-1} e^{-\phi u} \, du$ ,

which may be rewritten as

$\displaystyle \frac{1}{\phi^p} = \displaystyle \frac{1}{\Gamma(p)} \int_0^\infty t^{p-1} e^{-\phi t} \, dt$

after changing the dummy variable back to $t$ .

A simple (!) application of this method is the famous Dirichlet integral

$I = \displaystyle \int_0^\infty \frac{\sin x}{x} \, dx$

which is pretty much unsolvable using techniques from freshman calculus. However, by substituting $\phi = x$ and $p=1$ in the above gamma equation, and using the fact that $\Gamma(1) = 0! = 1$ , we obtain

$I = \displaystyle \int_0^\infty \sin x \int_0^\infty e^{-xt} \, dt \, dx$

$= \displaystyle \int_0^\infty \int_0^\infty e^{-xt} \sin x \, dx \, dt$

after interchanging the order of integration. The inner integral can be found by integration by parts and is often included in tables of integrals:

$I = \displaystyle \int_0^\infty -\left[ \frac{e^{-xt} (\cos x + t \sin x)}{1+t^2} \right]_{x=0}^{x=\infty} \, dt$

$= \displaystyle \int_0^\infty \left[0 +\frac{e^{0} (\cos 0 + t \sin 0)}{1+t^2} \right] \, dt$

$= \displaystyle \int_0^\infty \frac{1}{1+t^2} \, dt$ .

At this point, the integral is now a standard one from freshman calculus:

$I = \displaystyle \left[ \tan^{-1} t \right]_0^\infty = \displaystyle \frac{\pi}{2} - 0 = \displaystyle \frac{\pi}{2}$ .

In the article, the authors give many more applications of this method to other integrals, thus illustrating the famous quote, “An idea which can be used only once is a trick. If one can use it more than once it becomes a method.” The authors also add, “We present some examples to illustrate the utility of this technique in the hope that by doing so we may convince the reader that it makes a valuable addition to one’s integration toolkit.” I’m sold.

Integration Using Schwinger Parametrization

Published by John Quintanilla

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