Pirooz Mohazzabi^*, Nathan Hugh Jones
Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, Wisconsin.
DOI: 10.4236/jamp.2026.144075   PDF    HTML   XML   18 Downloads   274 Views   Citations

Abstract

A certain class of infinite sums leading to Riemann integrals that involve additional differentials inside their integrands is investigated. It is shown that these integrals can easily be evaluated by applying a simple algorithm.

Keywords

Infinite, Sum, Riemann, Integral, Integrand, Differential

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1. Introduction

There are situations where additional differential expressions appear inside the integrand of a definite integral, namely

${\displaystyle {\int }_a^{b}f\left(x,\text{d}x\right)\text{d}x}$ (1)

where $f\left(x,\text{d}x\right)$ is a function of both the variable $x$ and its differential $\text{d}x$ . Although these situations are extremely rare, they do happen in certain areas of mathematics, probability, information theory, and science [1].

Consider an interval $\left[a,b\right]$ of real numbers $x$ . The interval is divided into $n$ small subintervals, each of width $\text{Δ}{x}_i$ ($i=1,2,3,\cdots ,n$ ), as shown in Figure 1. Let the value of $x$ at the midpoint of slice $i$ be $x_i$ .

Suppose we have a function of the independent variables $x$ and its increment $\text{Δ}x$ , denoted by $f\left(x,\text{Δ}x\right)$ . Then for the interval $i$ , the value of this function is approximately $f\left(x_i,\text{Δ}{x}_i\right)$ . We construct the sum,

$S_n={\displaystyle \sum }_{i=1}^{n}f\left(x_i,\Delta x_i\right)\Delta x_i$ (2)

We now take the limit $n\to \infty$ such that each $\text{Δ}{x}_i\to 0$ ,

$S=\underset{n\to \infty }{\lim }{S}_n=\underset{n\to \infty }{\lim }{\displaystyle \sum }_{i=1}^{n}f\left(x_i,\Delta x_i\right)\Delta x_i$ (3)

Figure 1. An interval $\left[a,b\right]$ is divided into $n$ subintervals, each of width $\text{Δ}{x}_i$ . The coordinate $x_i$ is the midpoint of subinterval $i$ .

Sums and limits like these, although rare, appear in some areas such as information theory. For example, Shannon entropy $S$ for discrete probabilities is defined by [2]-[5]

$S=-{\displaystyle \sum }_{i=1}^n{p}_i\ln p_i$ (4)

where $p_i$ is the probability of the outcome $E_i$ in a random event with outcomes $E_1,E_2,\cdots ,E_n$ . Extension of this definition to a continuous random variable with a probability density function $g\left(x\right)$ results in the equation

$S_n=-{\displaystyle \sum }_{i=1}^{n}g\left(x_i\right)\Delta x_i\ln \left[g\left(x_i\right)\Delta x_i\right]=-{\displaystyle \sum }_{i=1}^{n}g\left(x_i\right)\ln \left[g\left(x_i\right)\Delta x_i\right]\Delta x_i$ (5)

where here the function $g\left(x_i\right)\ln \left[g\left(x_i\right)\Delta x_i\right]$ plays the role of $f\left(x_i,\Delta x_i\right)$ of Equation (3).

In what follows, we show that the limit of a sum like Equation (3) can be converted into a Riemann integral just as the limit of ordinary sums [6],

$\underset{n\to \infty }{\lim }{\displaystyle \sum }_{i=1}^{n}f\left(x_i\right)\Delta x_i={\displaystyle {\int }_a^{b}f\left(x\right)\text{d}x}$ (6)

except that for evaluation of the resulting integral, the $\text{d}x$ in the integrand should be set equal to zero, namely,

$\underset{n\to \infty }{\lim }{\displaystyle \sum }_{i=1}^{n}f\left(x_i,\Delta x_i\right)\Delta x_i={\displaystyle {\int }_a^{b}f{\left(x,\text{d}x\right)}_{\text{d}x=0}\text{d}x}$ (7)

where $\left[a,b\right]$ is the support set of the random variable $x$ . But, if the integrand becomes undefined at $\text{d}x=0$ , its limit should be considered for $\text{d}x\to 0$ . Note that this is not a new integration rule, it is simply Riemann integral, in which $\text{d}x$ is set equal to zero in its integrand.

2. Proof of Equation 7

Let us expand each function $f\left(x_i,\Delta x_i\right)$ of the sum in Equation (3) with respect to $\text{Δ}{x}_i$ in a Taylor series about $\text{Δ}{x}_i=0$ ,

$\begin{array}{c}S=\underset{n\to \infty }{\lim }{\displaystyle \sum }_{i=1}^n\left[f\left(x_i,0\right)+{\left(\frac{\partial f}{\partial \Delta x_i}\right)}_0\Delta x_i+\frac{1}{2}{\left(\frac{{\partial }^{2}f}{\partial \Delta x_i^2}\right)}_0\Delta x_i^2+\cdots \right]\Delta x_i\\ =\underset{n\to \infty }{\lim }{\displaystyle \sum }_{i=1}^n\left[f\left(x_i,0\right)\Delta x_i+{\left(\frac{\partial f}{\partial \Delta x_i}\right)}_0\Delta x_i^2+\frac{1}{2}{\left(\frac{{\partial }^{2}f}{\partial \Delta x_i^2}\right)}_0\Delta x_i^3+\cdots \right]\end{array}$ (8)

However, as each $\text{Δ}{x}_i\to 0$ , the second and higher powers of $\text{Δ}{x}_i$ become negligible compared to its first power, and we obtain

$S=\underset{n\to \infty }{\lim }{\displaystyle \sum }_{i=1}^{n}f\left(x_i,0\right)\Delta x_i$ (9)

But this, by definition, is the Riemann integral,

$S={\displaystyle {\int }_a^{b}f\left(x,0\right)\text{d}x}$ (10)

which, for more clarity, can be written as

$S={\displaystyle {\int }_a^{b}f{\left(x,\text{d}x\right)}_{\text{d}x=0}\text{d}x}$ (11)

which proves Equation (7).

3. Additional Examples

A simple example is the following sum over the interval $a=0$ and $b=1$ ,

$\begin{array}{c}\underset{n\to \infty }{\lim }{\displaystyle \sum }_{i=1}^n\sqrt{{\text{e}}^{x_i}\cos \left(\Delta x_i\right)}\Delta x_i={\displaystyle {\int }_0^1\sqrt{{\text{e}}^x\cos \left(\text{d}x\right)}\text{d}x}={\displaystyle {\int }_0^1\sqrt{{\text{e}}^x\cos \left(0\right)}\text{d}x}\\ ={\displaystyle {\int }_0^1{\text{e}}^{x/2}\text{d}x}=1.2974\end{array}$

Numerical evaluation of the left hand side of this equation verifies the result.

Evaluation of integrals involving $\text{d}x$ in the integrand should adhere to the rule explained above. Otherwise, incorrect answers may result. More specifically, the differential $\text{d}x$ within the integrand and that of the integral should be treated differently; the former should be set equal to zero while the latter should be considered as infinitesimal. For example, the integral

${\displaystyle {\int }_1^2\ln \left(\text{d}x\right)\text{d}x}$ (12)

may seem to be equal to zero because ${\lim }_{\text{d}x\to 0}\text{d}x\ln \left(\text{d}x\right)=0$ . However, this result is incorrect. The correct answer is

${\displaystyle {\int }_1^2\ln \left(\text{d}x\right)\text{d}x}={\displaystyle {\int }_1^2\ln \left(0\right)\text{d}x}=\ln \left(0\right){\displaystyle {\int }_1^2\text{d}x}=-\infty$ (13)

which can be verified by numerically evaluating the following sum over the indicated interval,

$\underset{n\to \infty }{\lim }{\displaystyle \sum }_{i=1}^n\ln \left(\Delta x_i\right)\Delta x_i$ (14)

4. Discussion and Conclusion

There are situations where the differential of a Riemann integral also appears in the integrand. This happens when the infinite sum whose limit yields the Riemann integral contains the term $\text{Δ}x$ in its function. However, these situations are highly uncommon. To the best of our knowledge, they are not mentioned in any mathematics textbooks, and a literature survey did not produce any results. In conclusion, because these cases have shown up occasionally in science and mathematics [1], they warrant an examination and explanation, which has been the objective of this article.

Returning to the example of Shannon entropy for a continuous random variable, taking the limit of Equation (5), we obtain

$S=-{\displaystyle {\int }_a^{b}g\left(x\right)\text{ln}{\left[g\left(x\right)\text{d}x\right]}_{\text{d}x=0}\text{d}x}=-\text{ln}\left(0\right){\displaystyle {\int }_a^{b}g\left(x\right)\text{d}x}=\infty$ (15)

Consequently, the continuous Shannon entropy, as a direct limit of the discrete entropy, diverges without renormalization. However, this renormalization alters the physical meaning of the resulting equation, and hence should no longer be called Shannon entropy [7]. Nevertheless, a different type of continuous entropy, or differential entropy, has been defined by [8]-[10]

$S=-{\displaystyle {\int }_a^{b}g\left(x\right)\ln g\left(x\right)\text{d}x}$ (16)

where the limits $a$ and $b$ define the support set, or the interval of the random variable $x$ , and $g\left(x\right)$ is the probability density function for $x$ . Despite its shortcomings, this entropy is applied in areas such as thermodynamics, statistical mechanics, and information theory [7].

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

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[4]	Rozanov, Y.A. (1969) Probability Theory: A Concise Course. Revised English Edition, Translated and Edited by R.A. Silverman, Dover, 117.
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