# Integration & Summation

Standard

A few months ago I wrote a series of blog posts on “rigorous”  definitions of integration [Part 1, Part 2]. Last week I identified an interesting flaw in my “imagination” of integration in terms of “limiting summation” and it lead to an interesting investigation.

While defining integration as area under curve, we consider rectangles of equal width and let that width approach zero. Hence I used to imagine integration as summation of individual heights, since width approaches zero in limiting case. It was just like extending summation over integers to summation over real numbers.

My Thought Process..

But as per my above imagination, since width of line segment is zero,  I am considering rectangles of zero width. Then each rectangle is of zero area (I proved it recently). So the area under curve will be zero! Paradox!

I realized that, just like ancient greeks, I am using very bad imagination of limiting process!

The Insight

But, as it turns out, my imagination is NOT completely wrong.  I googled and stumbled upon this stack exchange post:

There is the answer by Jonathan to this question which captures my imagination:

The idea is that since $\int_0^n f(x)dx$ can be approximated by the Riemann sum, thus $\displaystyle{\sum_{i=0}^n f(i) = \int_{0}^n f(x)dx + \text{higher order corrections}}$

The generalization of above idea gives us the Euler–Maclaurin formula

$\displaystyle{\sum_{i=m+1}^n f(i) = \int^n_m f(x)\,dx + B_1 \left(f(n) - f(m)\right) + \sum_{k=1}^p\frac{B_{2k}}{(2k)!}\left(f^{(2k - 1)}(n) - f^{(2k - 1)}(m)\right) + R}$

where $m,n,p$ are natural numbers, $f (x)$ is a real valued continuous function, $B_k$ are the Bernoulli numbers and $R$ is an error term which is normally small for suitable values of $p$ (depends on $n, m, p$ and $f$).

Proof of above formula is by principle of mathematical induction. For more details, see this beautiful paper: Apostol, T. M.. (1999). An Elementary View of Euler’s Summation Formula. The American Mathematical Monthly, 106(5), 409–418. http://doi.org/10.2307/2589145

# So many Integrals – II

Standard

As promised in previous post, now I will briefly discuss the remaining two flavors of Integrals.

Stieltjes Integral

Stieltjes

In 1894, a Dutch mathematician, Thomas Stieltjes, while solving the moment problem, that is, given the moments of all orders of a body, find the distribution of its mass, gave a generalization of the Darboux integral.

Let $P : a = x_0 < x_1 < x_2<\ldots < x_n = b$, $n$ being an integer, be a partition of the interval $[a, b]$.

For a function $\alpha$, monotonically increasing on $[a,b]$, we write:

$\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1})$

Let $f$ be a bounded function defined on an interval $[a, b],\quad a, b$ being real numbers. We define the sum

$S_P = \sum_{i=1}^n f(t_i)\Delta \alpha_i, \quad \overline{S}_P = \sum_{i=1}^n f(s_i)\Delta \alpha_i$

where $t_i,s_i \in [x_{i-1} , x_i]$ be such that

$f(t_i) = \text{sup} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$,

$f(s_i) = \text{inf} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$

If the $\text{inf}\{S_P\}$ and $\text{sup}\{\overline{S}_P\}$ are equal, we denote the common value by  $\int_{a}^{b} f(x) d\alpha(x)$ and call it Steiltjes integral of $f$ with respect to $\alpha$ over $[a,b]$.

Lebesgue Integral

Lebesgue

Let me quote Wikipedia article:

The integral of a function f between limits a and b can be interpreted as the area under the graph of f. This is easy to understand for familiar functions such as polynomials, but what does it mean for more exotic functions? In general, for which class of functions does “area under the curve” make sense? The answer to this question has great theoretical and practical importance.

In 1901, a French mathematician, Henri Léon Lebesgue generalized the notion of the integral by extending the concept of the area below a curve to include functions with uncountable discontinuities .

Lebesgue defined his integral by partitioning the range of a function and summing up sets of x-coordinates belonging to given y-coordinates, rather than, as had traditionally been done, partitioning the domain.

Lebesgue himself, according to his colleague, Paul Montel, compared his method with paying off a debt: (see:pp. 803,  The Princeton Companion to Mathematics)

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.

A set $\mathcal{A}$ is said to be Lebesgue measurable, if for each set $\mathcal{E} \subset \mathbb{R}$ the Carathéodory condition:

$m^{*} (\mathcal{E}) = m^{*}(\mathcal{E} \cap \mathcal{A}) + m^{*}(\mathcal{E}\backslash \mathcal{A})$

is satisfied, where $m^{*}(\mathcal{A})$ is called outer measure and is defined as:

$m^{*}(\mathcal{A}) = \inf\sum\limits_{n=1}^\infty (b_n-a_n)$

where $\mathcal{A}$ is a countable collection of closed intervals $[a_n,b_n], a_n\leq b_n$, that cover $\mathcal{A}$.

The Lebesgue integral of a simple function $\phi(x) = \sum_{i=1}^n c_i \chi_{\mathcal{A}_i} (x)$ on $\mathcal{A}$, where $\mathcal{A}=\bigcup_{i=1}^{\infty} \mathcal{A}_{i}$, $\mathcal{A}_i$ are pairwise disjoint measurable sets and $c_1, c_2, \ldots$ are real numbers, is defined as:

$\int\limits_{\mathcal{A}} \phi dm = \sum\limits_{i=1}^{n} c_i m(\mathcal{A}_i)$

where, $m(\mathcal{A}_i)$ is the Lebesgue measure of a measurable set $\mathcal{A}_i$.

An extended real value function $f: \mathcal{A}\rightarrow \overline{\mathbb{R}}$ defined on a measurable set $\mathcal{A}\subset\mathbb{R}$ is said to be Lebesgue measurable on $\mathcal{A}$ if $f^{-1} ((c,\infty]) = \{x \in\mathcal{A} : f(x) > c\}$ is a Lebesgue measurable subset of $\mathcal{A}$ for every real number $c$.

If $f$ is Lebesgue measurable and non-negative on $\mathcal{A}$ we define:

$\int\limits_{\mathcal{A}} f dm = \sup \int\limits_{\mathcal{A}} \phi dm$

where the supremum is taken over all simple functions $\phi$ such that $0\leq \phi \leq f$.

The function $f$ is said to be Lebesgue integrable on $\mathcal{A}$ if it’s integral over $\mathcal{A}$ is finite.

The Lebesgue integral is deficient in one respect. The Riemann integral generalizes to the improper Riemann integral to measure functions whose domain of definition is not a closed interval. The Lebesgue integral integrates many of these functions, but not all of them.

# So many Integrals – I

Standard

We all know that, area is  the basis of integration theory, just as counting is basis of the real number system. So, we can say:

An integral is a mathematical operator that can be interpreted as an area under curve.

But, in mathematics we have various flavors of integrals named after their discoverers. Since the topic is a bit long, I have divided it into two posts. In this and next post I will write their general form and then will briefly discuss them.

Cauchy Integral

Newton, Leibniz and Cauchy (left to right)

This was rigorous formulation of Newton’s & Leibniz’s idea of integration, in 1826 by French mathematician, Baron Augustin-Louis Cauchy.

Let $f$ be a positive continuous function defined on an interval $[a, b],\quad a, b$ being real numbers. Let $P : a = x_0 < x_1 < x_2<\ldots < x_n = b$, $n$ being an integer, be a partition of the interval $[a, b]$ and form the sum

$S_p = \sum_{i=1}^n (x_i - x_{i-1}) f(t_i)$

where $t_i \in [x_{i-1} , x_i]f$ be such that $f(t_i) = \text{Minimum} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$

By adding more points to the partition $P$, we can get a new partition, say $P'$, which we call a ‘refinement’ of $P$ and then form the sum $S_{P'}$.  It is trivial to see that $S_P \leq S_{P'} \leq \text{Area bounded between x-axis and function}f$

Since, $f$ is continuous (and positive), then $S_P$ becomes closer and closer to a unique real number, say $kf$, as we take more and more refined partitions in such a way that $|P| := \text{Maximum} \{x_i - x_{i-1}, 1 \leq i \leq n\}$ becomes closer to zero. Such a limit will be independent of the partitions. The number $k$ is the area bounded by function and x-axis and we call it the Cauchy integral of $f$ over $a$  to $b$. Symbolically, $\int_{a}^{b} f(x) dx$ (read as “integral of f(x)dx from a to b”).

Riemann Integral

Riemann

Cauchy’s definition of integral can readily be extended to a bounded function with finitely many discontinuities. Thus, Cauchy integral does not require either the assumption of continuity or any analytical expression of $f$ to prove that the sum $S_p$ indeed converges to a unique real number.

In 1851, a German mathematician, Georg Friedrich Bernhard Riemann gave a more general definition of integral.

Let $[a,b]$ be a closed interval in $\mathbb{R}$. A finite, ordered set of points $P :\{ a = x_0 < x_1 < x_2<\ldots < x_n = b\}$, $n$ being an integer, be a partition of the interval $[a, b]$. Let, $I_j$ denote the interval $[x_{j-1}, x_j], j= 1,2,3,\ldots , n$. The symbol $\Delta_j$ denotes the length of $I_j$. The mesh of $P$, denoted by $m(P)$, is defined to be $max\Delta_j$.

Now, let $f$ be a function defined on interval $[a,b]$. If, for each $j$, $s_j$ is an element of $I_j$, then we define:

$S_P = \sum_{j=1}^n f(s_j) \Delta_j$

Further, we say that $S_P$ tend to a limit $k$ as $m(P)$ tends to 0 if, for any $\epsilon > 0$, there is a $\delta >0$ such that, if $P$ is any partition of $[a,b]$ with $m(P) < \delta$, then $|S_P - k| < \epsilon$ for every choice of $s_j \in I_j$.

Now, if $S_P$ tends to a finite limit as $m(P)$ tends to zero, the value of the limit is called Riemann integral of $f$ over $[a,b]$ and is denoted by $\int_{a}^{b} f(x) dx$

Darboux Integral

Darboux

In 1875, a French mathematician, Jean Gaston Darboux  gave his way of looking at the Riemann integral, defining upper and lower sums and defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller.

Let $f$ be a bounded function defined on an interval $[a, b],\quad a, b$ being real numbers. Let $P : a = x_0 < x_1 < x_2<\ldots < x_n = b$, $n$ being an integer, be a partition of the interval $[a, b]$ and form the sum

$S_P = \sum_{i=1}^n (x_i - x_{i-1}) f(t_i), \quad \overline{S}_P =\sum_{i=1}^n (x_i - x_{i-1}) f(s_i)$

where $t_i,s_i \in [x_{i-1} , x_i]$ be such that

$f(t_i) = \text{sup} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$,

$f(s_i) = \text{inf} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$

The sums $S_P$ and $\overline{S}_P$ represent the areas and  $S_P \leq \text{Area bounded by curve} \leq \overline{S}_P$. Moreover, if $P'$ is a refinement of $P$, then

$S_p \leq S_{P'} \leq \text{Area bounded by curve} \leq \overline{S}_{P'} \leq \overline{S}_{P}$

Using the boundedness of $f$, one can show that $S_P, \overline{S}_P$ converge as the partition get’s finer and finer, that is $|P| := \text{Maximum}\{x_i - x_{i-1}, 1 \leq i \leq n\} \rightarrow 0$, to some real numbers, say $k_1, k_2$ respectively. Then:

$k_l \leq \text{Area bounnded by the curve} \leq k_2$

If $k_l = k_2$ , then we have $\int_{a}^{b} f(x) dx = k_l = k_2$.

There are two more flavours of integrals which I will discuss in next post. (namely, Stieltjes Integral and Lebesgue Integral)