The revelation of beautiful structures innate to dimensionality itself
Sake of Its Beauty
Mathematics is a natural expression that unwinds the beautiful structures innate to dimensionality itself. Mathematics is eternal, its structures exist either expressed or unexpressed in the latent fabric of the universe. Many people think that mathematical ideas are static. They think that the ideas originated at some time in the historical past and remain unchanged for all future times. But the one who knows mathematics from the inside, the subject has rather than feeling of living thing. It’s more than just discipline for calculation and application. Mathematics and related fields of study grow daily through the accretion of new information, it changes daily by regarding itself and the world from new vantage points, it maintains a regulatory balance by consisting to the oblivion of irrelevancy a fraction of its past accomplishments.
The purpose of this article is to illustrate the process of growth of the most mysterious and fascinating fact introduced by mathematician Leonhard Euler’s ‘Eulerian Integral’. This growth partook of the general development of mathematics over the past two and a quarter centuries. Of the so-called “higher mathematical functions,” the gamma function is undoubtedly the most fundamental. It is simple enough for juniors in college to meet but deep enough to have called forth contributions from the finest mathematicians. And it is sufficiently compact to allow its profile to be sketched within the space of a brief essay.
Old is Gold
A New Horizon to Mathematics
The year 1729 saw the birth of the gamma function in correspondence between a Swiss mathematician in St. Petersburg and a German mathematician in Moscow. The former: Leonhard Euler (1707-1783), then 22 years of age, but became a prodigious mathematician, the greatest of the 18th century. The latter: Christian Goldbach (1690-1764), a savant, a man of many talents, and in correspondence with the leading thinkers of the day. As a mathematician, he was something of a dilettante, yet he was a man who bequeathed to the future a problem in the theory of numbers so easy to state and so difficult to prove that even to this day it remains on the mathematical horizon as a challenge.
All Intents And Purposes
The birth of the gamma function was due to the merging of several mathematical streams. The first was that of interpolation theory, a very practical subject largely the product of English mathematicians of the 17th century but which all mathematicians enjoyed dipping into from time to time. The second stream was that of the integral calculus and of the systematic building up of the formulas of indefinite integration, a process that had been going on steadily for many years. A certain ostensibly simple problem of interpolation arose and was bandied about unsuccessfully by Goldbach and by Daniel Bernoulli (1700- 1784) and even earlier by James Stirling (1692-1770). The problem was posed to Euler. Euler announced his solution to Goldbach in two letters which were to be the beginning of an extensive correspondence that lasted the duration of Goldbach’s life. The first letter dated October 13, 1729, dealt with the interpolation problem, while the second dated January 8, 1730, dealt with integration and tied the two together. Euler wrote Goldbach the merest outline, but within a year he published all the details in an article
De progressionibus transcendent- ibus seu quarum termini generales algebraice dari nequeunt.
Mathematical Justifications
This article can now be found reprinted in Volume \(\text{I}_{14}\) of Euler’s Opera Omnia. Since the interpolation problem is the easier one, let us begin with it.
\(\text{One of the simplest sequences of the integers which leads to an interesting theory is}\) \(1,1+2,1+2+3,1+2+3+4,\ldots\) \(\text{These are the triangular numbers,}\) \(\text{so-called}\) \(\text{because they}\) \(\text{represent the number of objects which can be placed}\) \(\text{in a triangular}\) \(\text{array of various sizes.}\)
Call the \(n^{\text{th}}\) one \(T_n ; \text{where} \; T_n = \frac{1}{2}n(n+1).\) Precisely, this formula simplifies computation by reducing a large number of additions to three fixed operations: one of addition, one of multiplication, and one of division. Thus, instead of adding the first hundred integers to obtain \(T_{100}\), we can compute \(T_{100}=\frac{1}{2}(100).(100+1) = 5050.\) Secondly, even though it doesn’t make literal sense to ask for, say, the sum of the first \(5\frac{1}{2}\) integers, the formula for \(T_n\) produces an answer to this. For whatever it is worth, the formula yields \(T_{5\frac{1}{2}}=\frac{1}{2}(5\frac{1}{2})(5\frac{1}{2}+1)=17\frac{7}{8}\). In this way, the formula extends the scope of the original problem to values of the variable other than those for which it was originally defined and solves the problem of interpolating between the known elementary values.This type of question, one which asks for an extension of meaning, cropped up frequently in the \(17 ^{\text{th}}\) and \(18^{\text{th}}\) centuries. Consider, for instance, the algebra of exponents.
The quantity \(a^m\) is defined initially as the product of \(m\) successive \(a\)’s. This definition has meaning when \(m\) is a positive integer, but what would \(a^{5\frac{1}{2}}\) it be? The product of \(5\frac{1}{2}\) successive \(a\)’s? The mysterious definitions \(a^0=1,a^{\frac{m}{n}}=\sqrt[n]{a^m},a^{-m}=\frac{1}{a^m}\), which solve this enigma and which are employed so fruitfully in algebra were written down explicitly for the first time by Newton in 1676. They are justified by a utility that derives from the fact that the definition leads to continuous exponential functions and that the law of exponents \(a^m.a^n=a^{m+n}\) becomes meaningful for all exponents whether positive integers or not.
An Interpolation Problem
Other problems of this type proved harder. Thus, Leibnitz introduced the notation \(d^n\) for the \(n^{\text{th}}\) iterate of the operation of differentiation. Moreover, he identified \(d^{-1}\) with \(\int\) and \(d^{-n}\) with the iterated integral. Then he tried to breathe some sense into the symbol \(d^{n}\) when \(n\) is any real value whatever. What, indeed, is the \(5\frac{1}{2}\) derivative of a function? This question had to wait for almost two centuries for a satisfactory answer.
\(\text{THE FACTORIALS}\)
| \(n\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\ldots\) |
| \(n!\) | \(1\) | \(2\) | \(6\) | \(24\) | \(120\) | \(720\) | \(5040\ldots\) |
\(\text{Question: What number should be inserted in the lower line half way between}\) \(\text{the}\) \(\text{upper}\; 5\; \text{and}\; 6\;?\)
\(\text{Euler's Answer: } 287.8852\ldots\) . \(\text{Hadamard's Answer: } 280.3002\ldots\)
But to return to our sequence of triangular numbers. If we change the plus signs to multiplication signs we obtain a new sequence: \(1, 1.2, 1.2.3,\ldots\) .This is the sequence of factorials. The factorials are usually abbreviated \(1!, 2!, 3!,\ldots\) and the first five are \(1, 2, 6, 24, 120\). They grow in size very rapidly. The number \(100!\) if written out in full would have \(158\) digits. By contrast, \(T_{100}=5050\) has a mere four digits. Factorials are omnipresent in mathematics; one can hardly open a page of mathematical analysis without finding it strewn with them. This being the case, is it possible to obtain an easy formula for computing the factorials? And is it possible to interpolate between the factorials? What should \(5\frac{1}{2}\) be? This is the interpolation problem that led to the gamma function, the interpolation problem of Stirling, Bernoulli and Goldbach. As we know, these two problems are related, for when one has a formula there is the possibility of inserting intermediate values into it. And now comes the surprising thing. There is no there can be, no formula for the factorials which is of the simple type found for \(T\). This is implicit in the very title Euler chose for his article. Translate the Latin and we have On transcendental progressions whose general term cannot be expressed algebraically. The solution to factorial interpolation lay deeper than “mere algebra.” Infinite processes were required.
Introducing With a Special One
It is difficult to chronicle the exact course of scientific discovery. This is particularly true in mathematics where one traditionally omits from articles and books all accounts of false starts, of the initial years of bungling, and where one may develop one’s topic forward or backward or sideways to heighten the dramatic effect. As one distinguished mathematician put it, a mathematical result must appear straight from the heavens as a deus ex machina for students to verify and accept but not to comprehend. Euler, experimenting with infinite products of numbers, chanced to notice that if \(n\) is a positive integer, \[ \boxed{\bigg[\bigg(\dfrac{2}{1}\bigg)^n . \dfrac{1}{n+1}\bigg] \bigg[\bigg(\dfrac{3}{2}\bigg)^n . \dfrac{2}{n+2}\bigg] \bigg[\bigg(\dfrac{4}{3}\bigg)^n . \dfrac{3}{n+3}\bigg] \ldots=n!} \hspace{0.5cm} (1) \]Leaving aside all delicate questions as to the convergence of the infinite product, the reader can verify this equation by canceling out all the common factors which appear at the top and bottom of the left-hand side. Moreover, the left-hand side is defined (at least formally) for all kinds of \(n\) other than negative integers. Euler noticed also that when the value \(n = \frac{1}{2}\) is inserted, the left-hand side yields (after a bit of manipulation) the famous infinite product of the Englishman John Wallis (1616-1703):
\[\boxed{\bigg(\dfrac{2.2}{1.3}\bigg) \bigg(\dfrac{4.4}{3.5}\bigg) \bigg(\dfrac{6.6}{5.7}\bigg) \bigg(\dfrac{8.8}{7.9}\bigg) \ldots=\dfrac{\pi}{2}}\hspace{0.45 cm}\;(2) \]With this discovery, Euler could have stopped. His problem was solved. Indeed, the whole theory of the gamma function can be based on the infinite product \((1)\) which today is written more conventionally as,
\[ \boxed{\lim_{m \to \infty} \dfrac{m!(m+1)^n}{(n+1)(n+2)\ldots(n+m)}}\hspace{0.6 cm} (3) \]However, he went on. He observed that his product displayed the following curious phenomenon: for some values of \(n\), namely integers, it yielded integers, whereas, for another value, namely \(n = \frac{1}{2}\), it yielded an expression involving \(T\). Now meant circles and their quadrature, and quadratures meant integrals, and he was familiar with integrals that exhibited the same phenomenon. It, therefore, occurred to him to look for a transformation that would allow him to express his product as an integral.
He took up the integral \(\int_0^1 x^e(1-x)^n \ dx\). Special cases of it had already been discussed by Wallis, by Newton, and by Stirling. It was a troublesome integral to handle, for the indefinite integral is not always an elementary function of \(x\). Assuming that \(n\) is an integer, but that \(e\) is an arbitrary value, Euler expanded \((1-x)^n\) by the binomial theorem, and without difficulty found that
\[ \boxed{\int_0^1 x^e(1-x)^n\ dx = \dfrac{1.2\ldots n}{(e+1)(e+2)\ldots(e+n+1)}} \hspace{0.7568 cm} (4) \]Euler’s idea was to isolate the \(1.2.3\ldots n\) from the denominator so that he would have an expression for \(n!\) as an integral. He proceeds in this way.
\(\text{Here we follow Euler's own formulation and nomenclature, marking with an}\) \(\text{* those formulas which occur in the original paper. Euler wrote a plain}\; \int\; \text{for}\; \int_0^1.\) \(\text{He}\) \(\text{substituted}\) \(\frac{f}{g}\) \(\text{for}\) \(e\) \(\text{and found}\)
\[ \boxed{\int_0^1 x^{\frac{f}{g}}(1-x)^n\ dx = \dfrac{g^{n+1}}{f+(n+1)g}. \dfrac{1.2\ldots n}{(f+g)(f+2.g)\ldots(f+n.g)}} \hspace{0.35 cm} (5) \]
\(\text{And so,}\)
\[ \boxed{\dfrac{1.2\ldots n}{(f+g)(f+2.g)\ldots(f+n.g)}= \dfrac{f+(n+1)g}{g^{n+1}}\int x^{\frac{f}{g}}(1-x)^ndx\;}\;\;\; (6)^{*} \]He observed that he could isolate the \(1.2.3\ldots n\) if he set \(f=1\) and \(g=0\) in the lefthand member, but that if he did so, he would obtain on the right an indeterminate form which he writes quaintly as,
\[ \boxed{\int \dfrac{x^{\frac{1}{0}}(1-x)^n}{0^{n+1}}dx} \hspace{0.8cm}(7)^{*} \]He now proceeds to find the value of the expression \((7)^*\). He first made the substitution \(x^{\frac{g}{f+g}}\) in place of \(x\). This gave him
\[\boxed{\dfrac{g}{f+g}x^{-\frac{g}{f+g}}dx\;} \hspace{0.67 cm} (8)^*\] in place of \(dx\) and hence, the right-hand member of \((6)^*\) becomes,
\[ \boxed{\dfrac{f+(n+1)g}{g^{n+1}}\int\dfrac{g}{f+g}\bigg(1-x^{\frac{g}{f+g}}\bigg)^ndx \;}\hspace{0.6 cm} (9)^* \]Once again, Euler made a trial setting of \(f=1, g=0\) having presumably reduced this integral goes to
\[ \boxed{\dfrac{f+(n+1)g}{(f+g)^{n+1}} \int_0^1 \bigg(\dfrac{1-x^{\frac{g}{f+g}}}{\frac{g}{f+g}}\bigg)dx\;} \hspace{0.56 cm}(10) \]
and this yielded the indeterminate,
\[ \boxed{\int\dfrac{(1-x^0)^n}{0^n}dx\;} \hspace{0.45 cm} (11)^* \]
He now considered the related expression \(\frac{1-x^z}{z}\), for vanishing \(z\). He differentiates the numerator and denominator, as he says, by a known (\(\text{l'Hospital}\)) rule and obtained
\[\boxed{-\dfrac{x^zdzlx}{dz}} \hspace{0.67 cm} (12)^*\]
which for \(z=0\) produced \(-lx\). Thus,
\[\boxed{\dfrac{1-x^0}{0}=-lx\;} \hspace{0.87 cm} (13)^*\]
and
\[ \boxed{\dfrac{(1-x^0)^n}{0^n}=(-lx)^n\;} \hspace{0.87 cm} (14)^* \]
He, therefore, concluded that
\[ \boxed{n! = \int_0^1 (-\log x)^n dx} \hspace{0.87 cm} (15) \]This gave him what he wanted, an expression for \(n!\) as an integral wherein values other than positive integers may be substituted. The reader is encouraged to formulate his own criticism of Euler’s derivation. Students in advanced calculus generally meet Euler’s integral first in the form
\[ \boxed{\Gamma(x) = \int_0^{\infty} e^{-t}t^{x-1}dt} \ , e=2.71828\ldots \hspace{0.67 cm} (16) \]This modification of the integral \((15)\) as well as the Greek \(\Gamma\) is due to Adrien Marie Legendre (1752-1833). Legendre calls the integral \((4)\) with which Euler started his derivation the first Eulerian integral and \((15)\) the second Eulerian integral. The first Eulerian integral is currently known as the Beta function and is now conventionally written,
\[ \boxed{B(m,n)=\int_0^1 x^{m-1}(1-x)^{n-1}dx\;} \hspace{0.89 cm} (17) \]With the tools available in advanced calculus, it is readily established (how easily the great achievements of the past seem to be comprehended and duplicated!) that the integral possesses meaning when \(x>0\) and thus yields a certain function \(\Gamma(x)\) defined for these values. Moreover,
\[ \boxed {\Gamma(n+1) = n!}\;\;\; (18) \]whenever \(n\) is a positive integer. * It is further established that for all \(x>0\)
\[ x\Gamma(x) = \Gamma(x+1) \]This is the so-called recurrence relation for the gamma function and in the years following Euler it plays, as we shall see, an increasingly important role in its theory. These facts, plus perhaps the relationship between Euler’s two types of integrals
\[ B(m, n) =\dfrac{\Gamma(m)\Gamma(n)}{\Gamma(m +n)} \]
Expressio Analytica
In order to appreciate a little better the problem confronting Euler it is useful to skip ahead a bit and formulate it in an up-to-date fashion: find a reasonably simple function which at the integers \(1, 2, 3,…\) takes on the factorial values \(1, 2, 6,…\). Now today, a function is a relationship between two sets of numbers wherein to a number of one set is assigned a number of the second set. What is stressed is the relationship and not the nature of the rules which serve to determine the relationship. To help students visualize the function concept in its full generality, mathematics instructors are accustomed to draw a curve full of twists and discontinuities. The more of these the more general the function is supposed to be. Given, then, the points \((1,1), (2, 2), (3, 6), (4, 24),…\) and adopting the point of view wherein “function” is what we have just said, the problem of interpolation is one of finding a curve which passes through the given points. This is ridiculously easy to solve. It can be done in an unlimited number of ways. Merely take a pencil and draw some \(-\) curve any curve will do \(-\) which passes through the points. Such a curve automatically defines a function which solves the interpolation problem. In this way, too free an attitude as to what constitutes a function solves the problem trivially and would enrich mathematics but little. Euler’s task was different. In the early 18th century, a function was more or less synonymous with a formula, and by a formula was meant an expression which could be derived from elementary manipulations with addition, subtraction, multiplication, division, powers, roots, exponentials, logarithms, differentiation, integration, infinite series, i.e., one which came from the ordinary processes of mathematical analysis. Such a formula was called an expressio analytica, an analytical expression. Three important facts were now known: Euler’s integral, Euler’s product, and the functional or recurrence relationship \(x\Gamma(x)=\Gamma(x+1), x>0\). This last is the generalization of the obvious arithmetic fact that for positive integers, \((n+1)n!=(n+1)!\) It is a particularly useful relationship inasmuch as it enables us by applying it over and over again to reduce the problem of evaluating a factorial of an arbitrary real number whole or otherwise to the problem of evaluating the factorial of an appropriate number lying between \(0\) and \(1\). Thus, if we write \(n=4\frac{1}{2}\) in the above formula we obtain \((4\frac{1}{2}+1)!=5\frac{1}{2}(4\frac{1}{2})!\) If we could only find out what \((4\frac{1}{2})!\) is, then we would know that \((5\frac{1}{2})!\) is. This process of reduction to lower numbers can be kept up and yields
\[ \bigg(5\frac{1}{2}\bigg)! = \bigg(\dfrac{3}{2}\bigg) \bigg(\dfrac{5}{2}\bigg) \bigg(\dfrac{7}{2}\bigg) \bigg(\dfrac{9}{2}\bigg) \bigg(\dfrac{11}{2}\bigg) \bigg(\dfrac{1}{2}\bigg)! \]
and since we have \((\frac{1}{2})!=\frac{1}{2}\sqrt{\pi}\) from \((1)\) and \((2)\), we can now compute our answer. Such a device is obviously very important for anyone who must do calculations with the gamma function. Other information is forthcoming from the recurrence relationship. Though the formula \((n+1)n! =(n+1)!\) as a condensation of the arithmetic identity \((n+1).1.2\ldots n=1.2\ldots n.(n+1)\) makes sense only for \(n=1, 2, \text{etc}.\), blind insertions of other values produce interesting things. Thus, inserting \(n=0\), we obtain \(0!= 1\).
Building Blocks
Euler’s task was to find, if he could, an analytical expression arising naturally from the corpus of mathematics which would yield factorials when a positive integer was inserted, but which would still be meaningful for other values of the variable. Functions are the building blocks of mathematical analysis. In the 18th and 19th centuries mathematicians devoted much time and loving care to developing the properties and interrelationships between special functions. Powers, roots, algebraic functions, trigonometric functions, exponential functions, loga- rithmic functions, the gamma function, the beta function, the hypergeometric function, the elliptic functions, the theta function, the Bessel function, the Matheiu function, the Weber function, Struve function, the Airy function, Lamé functions, literally hundreds of special functions were singled out for scrutiny and their main features were drawn. This is an art which is not much cultivated these days. Times have changed and emphasis has shifted. Mathematicians on the whole prefer more abstract fare. Large classes of functions are studied instead of individual ones. Sociology has replaced biography. The field of special functions, as it is now known, is left largely to a small but ardent group of enthusiasts plus those whose work in physics or engineering confronts them directly with the necessity of dealing with such matters.
\(\text{The Euler gamma function is the only function defined for}\) \(x>0\) \(\text{which is}\) \(\text{positive, is}\) \(1\) \(\text{at}\) \(x=1,\) \(\text{satisfies the functional equation}\) \(x\Gamma(x)=\Gamma(x+1),\) \(\text{and is}\) \(\text{logarithmically}\) \(\text{convex.}\)
This theorem is at once so striking and so satisfying that the contemporary synod of abstractionists who write mathematical canon under the pen name of N. Bourbaki has adopted it as the starting point for its exposition of the gamma function. The proof: one page; the discovery: 193 years. There is much that we know about the gamma function. Since Euler’s day more than 400 major papers relating to it have been written. But a few things. remain that we do not know and that we would like to know. Perhaps the hardest of the unsolved problems deal with questions of rationality and transcendentality. Consider, for instance, the number \(\gamma=0.57721\). This is the Euler-Mascheroni constant. Though the numerical value of \(\gamma\) is known to hundreds of decimal places, it is not known at the time of writing whether \(\gamma\) is or is not a rational number. Another problem of this sort deals with the values of the gamma function itself. Though, curiously enough, the product \(\frac{\Gamma(1/4)}{\sqrt[4]{\pi}}\) can be proved to be transcendental, it is not known whether \(\Gamma(1/4)\) is even rational.
Conclusion
George Gamow, the distinguished physicist, quotes Laplace as saying that when the known areas of a subject expand, so also do its frontiers. Laplace evidently had in mind the picture of a circle expanding in an infinite plane. Gamow disputes this for physics and has in mind the picture of a circle expanding on a spherical surface. As the circle expands, its boundary first expands, but later contracts. This writer agrees with Gamow as far as mathematics is concerned. Yet the record is this: each generation has found something of interest to say about the gamma function. Perhaps the next generation will also.
See Also
References:
1. E. Artin, Einführung in die Theorie der Gammafunktion, Leipzig, 1931.
2. N. Bourbaki, Éléments de Mathématique, Book IV, Ch. VII, La Fonction Gamma, Paris,1951.
3. H. T. Davis, Tables of the Higher Mathematical Functions, vol. I, Bloomington, Indiana,1933.
4. L. Euler, Opera omnia, vol. I, Leipzig-Berlin, 1924.
5. P. H. Fuss, Ed., Correspondance Mathématique et Physique de Quelques Célèbres Geómètres du XVIIIeme Siècle, Tome I, St. Petersbourg, 1843.
6. G. H. Hardy, Divergent Series, Oxford, 1949, Ch. II.
7. F. Lösch and F. Schoblik, Die Fakultät und verwandte Funktionen, Leipzig, 1951.
8. N. Nielsen, Handbuch der Theorie der Gammafunktion, Leipzig, 1906.
9 . Table of the Gamma Function for Complex Arguments, National Bureau of Standards, Applied Math. Ser. 34, Washington, 1954.(Introduction by Herbert E. Salzer.
10. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge, 1947, Ch.12.