Riemann Integrals Containing Differentials within the Integrand

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.

• Infinite
• Sum
• Riemann
• Integral
• Integrand
• Differential

1. Introduction

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

(1) ∫ a b f ( x , d x ) d x

where f ( x , d x ) is a function of both the variable x and its differential 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 [ a , b ] of real numbers x . The interval is divided into n small subintervals, each of width Δ x i ( i = 1 , 2 , 3 , ⋯ , 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 Δ x , denoted by f ( x , Δ x ) . Then for the interval i , the value of this function is approximately f ( x i , Δ x i ) . We construct the sum,

(2) S n = ∑ i = 1 n   f ( x i , Δ x i ) Δ x i

We now take the limit n → ∞ such that each Δ x i → 0 ,

(3) S = lim n → ∞ S n = lim n → ∞ ∑ i = 1 n   f ( x i , Δ x i ) Δ x i Figure 1

Figure 1. An interval [ a , b ] is divided into n subintervals, each of width 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]

(4) S = − ∑ i = 1 n   p i ln p i

where p i is the probability of the outcome E i in a random event with outcomes E 1 , E 2 , ⋯ , E n . Extension of this definition to a continuous random variable with a probability density function g ( x ) results in the equation

(5) S n = − ∑ i = 1 n   g ( x i ) Δ x i ln [ g ( x i ) Δ x i ] = − ∑ i = 1 n   g ( x i ) ln [ g ( x i ) Δ x i ] Δ x i

where here the function g ( x i ) ln [ g ( x i ) Δ x i ] plays the role of f ( x i , Δ x i ) 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],

(6) lim n → ∞ ∑ i = 1 n   f ( x i ) Δ x i = ∫ a b f ( x ) d x

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

(7) lim n → ∞ ∑ i = 1 n   f ( x i , Δ x i ) Δ x i = ∫ a b f ( x , d x ) d x = 0 d x

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

2. Proof of Equation 7

Let us expand each function f ( x i , Δ x i ) of the sum in Equation (3) with respect to Δ x i in a Taylor series about Δ x i = 0 ,

(8) S = lim n → ∞ ∑ i = 1 n [ f ( x i , 0 ) + ( ∂ f ∂ Δ x i ) 0 Δ x i + 1 2 ( ∂ 2 f ∂ Δ x i 2 ) 0 Δ x i 2 + ⋯ ] Δ x i = lim n → ∞ ∑ i = 1 n [ f ( x i , 0 ) Δ x i + ( ∂ f ∂ Δ x i ) 0 Δ x i 2 + 1 2 ( ∂ 2 f ∂ Δ x i 2 ) 0 Δ x i 3 + ⋯ ]

However, as each Δ x i → 0 , the second and higher powers of Δ x i become negligible compared to its first power, and we obtain

(9) S = lim n → ∞ ∑ i = 1 n   f ( x i , 0 ) Δ x i

But this, by definition, is the Riemann integral,

(10) S = ∫ a b f ( x , 0 ) d x

which, for more clarity, can be written as

(11) S = ∫ a b f ( x , d x ) d x = 0 d x

which proves Equation (7).

3. Additional Examples

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

lim n → ∞ ∑ i = 1 n e x i cos ( Δ x i )   Δ x i = ∫ 0 1 e x cos ( d x )   d x = ∫ 0 1 e x cos ( 0 )   d x = ∫ 0 1 e x / 2 d x = 1.2974

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

Evaluation of integrals involving d x in the integrand should adhere to the rule explained above. Otherwise, incorrect answers may result. More specifically, the differential 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

(12) ∫ 1 2 ln ( d x ) d x

may seem to be equal to zero because lim d x → 0 d x ln ( d x ) = 0 . However, this result is incorrect. The correct answer is

(13) ∫ 1 2 ln ( d x ) d x = ∫ 1 2 ln ( 0 ) d x = ln ( 0 ) ∫ 1 2 d x = − ∞

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

(14) lim n → ∞ ∑ i = 1 n ln ( Δ x i ) Δ x i

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 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

(15) S = − ∫ a b g ( x ) ln [ g ( x ) d x ] d x = 0 d x = − ln ( 0 ) ∫ a b g ( x ) d x = ∞

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]

(16) S = − ∫ a b g ( x ) ln g ( x ) d x

where the limits a and b define the support set, or the interval of the random variable x , and g ( x ) 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].

References

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