“The problem of missing terms in term by term integration involving divergent integrals” E. A. Galapon (2016) arXiv:1606.09382

$termbyterm2$
Term by term integration is one of the frequently used methods in evaluating integrals and in constructing asymptotic expansions of integrals. It typically involves expanding the integrand or a factor of it and then interchanging the order of summation and integration. Sometimes this formal manipulation leads to an infinite series involving divergent integrals. This calls for an interpretation of the non-existent integrals, in particular, in assigning meaningful values to them.

There is no scarcity of the means of assigning values to divergent integrals. We mention dimensional regularization, analytic continuation, Abel limits, distributional approach, and finite-part integrals. So interpreting divergent integrals is not really a problem. The problem is that, when we assign any of the possible values, we typically obtain a result that reproduces some terms of the exact result but completely misses out groups of terms.

Let us give an example how naive term by term integration with divergent integrals leads to missing terms. Let us consider the integral

(1) $\begin{equation*} \int_0^a \frac{\mathrm{d}x}{\lambda + x}=-\ln\lambda + \ln(\lambda + a), \;\;\; \lambda, a>0 . \end{equation*}$

Let us binomially expand $(\lambda+ x)^{-1}$ as follows

(2) $\begin{equation*} \frac{1}{\lambda + x} = \sum_{j=0}^{\infty} (-1)^j \frac{\lambda^j}{x^{j+1}}, \end{equation*}$

and then substitute the expansion back in the left hand side of equation \eqref{integral}, followed by term by term integration. The whole process yields the formal infinite series

(3) $\begin{equation*} \int_0^a \frac{\mathrm{d}x}{\lambda + x} \leadsto \sum_{j=0}^{\infty} (-1)^j \lambda^j \int_0^a \frac{1}{x^{j+1}} \mathrm{d}x . \end{equation*}$

Notice that the integrals are divergent, which requires us that we decide on how to assign values to them.

We can interpret them as finite part integrals (FPI). The finite part of the divergent integrals are obtained as follows. For a given integer $j=0,1,2,\dots$ , let $\epsilon>0$ and consider the integral

(4) $\begin{equation*} \int_{\epsilon}^{a} \frac{\mathrm{d}x}{x^{j+1}} = \left\{\begin{array}{ll} \ln a - \ln \epsilon &, \;\; j=0 \\ -\frac{1}{j}\left(\frac{1}{a^j}- \frac{1}{\epsilon^{j}}\right) &, \;\; j=1,2,\dots \end{array} \right. . \end{equation*}$

The finite part of the divergent integral $\int_0^ax^{-j-1} \mathrm{d}x$ is the term in equation \eqref{divergent} that has a finite limit as $\epsilon\rightarrow 0$ . In other words, the finite part is obtained by just simply dropping all terms in equation \eqref{divergent} that diverge in the said limit. We then obtain

(5) $\begin{equation*} \mathrm{FPI}\int_0^a \frac{\mathrm{d}x}{x^{j+1}} =\left\{ \begin{array}{ll} \ln a &,\;\; j=0 \\ -\frac{1}{j a^j} &, \;\;j=1, 2, \dots \end{array} \right. \end{equation*}$

We substitute these finite-part integrals back into the formal infinite series \eqref{infinite} to yield

(6) $\begin{equation*} \ln a + \sum_{s=1}^{\infty} \frac{(-1)^{s+1}}{s} \frac{\lambda^s}{a^s} \end{equation*}$

Now we expand the right hand side of equation \eqref{integral} for sufficiently large $a$ , and obtain

(7) $\begin{equation*} \int_0^a \frac{\mathrm{d}x}{\lambda+ x}=-\ln\lambda + \ln a + \sum_{s=1}^{\infty} \frac{(-1)^{s+1}}{s} \frac{\lambda^s}{a^s} . \end{equation*}$

Comparing the two expansions \eqref{formal} and \eqref{exact}, we find them to agree except on the first term in the second expansion. This demonstrates how naive term by term integration and careless interpretation of divergent integrals can lead to missing terms.

In this preprint, “The problem of missing terms in term by term integration involving divergent integrals”, I offer a solution to the missing term problem for the Stieltjes transform by lifting the integration in the complex plane. There it is shown that the missing terms arise from the singularities of the integrand, with the divergent integrals arising from term by term integration interpreted as finite-part integrals.

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