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e: The Master of All BRIAN J. MCCARTIN

Read Euler, read Euler. He is the master of us all. --P. S. de Laplace

aplace's dictum may rightfully be transcribed as: "Study Thus, e was first conceived as the limit e, study e. It is the master of all." L e = lim (1 + l/n)*' = 2.718281828459045" (1) Just as Gauss earned the moniker Princeps mathemati- n~ao corum amongst his contemporaries [1], e may be dubbed although its baptism awaited Euler in the eighteenth cen- Princeps constantium symbolum. Certainly, Euler would tury [2]. One might naively expect that all that could be concur, or why would he have endowed it with his own gleaned from equation (1) would have been mined long initial [2]? ago. Yet, it was only very recently that the asymptotic The historical roots of e have been exhaustively traced development and are readily available [3, 4, 5]. Likewise, certain fundamental properties of e such as its limit definition, its series =(1+1/n) ~ = ~ e~. representation, its close association with the rectangular hyperbola, and its relation to compound interest, radioactive decay, and the trigonometric and hyperbolic functions are ev= e l---7--. ' (2) k=o (v+ k)! l=o too well known to warrant treatment here [6, 7]. Rather, the focus of the present article is on matters re- with & denoting the Stirling numbers of the first kind [9], lated to e which are not so widely appreciated or at least was discovered [10]. have never been housed under one roof. When I do in- Although equation (1) is traditionally taken as the defin- dulge in reviewing well-known facts concerning e, it is to ition of e, it is much better approximated by the limit forge a link to results of a more exotic variety. [ (n + 2)n+2 (n + l)n+l ] Occurrences of e throughout pure and applied mathe- e = n--+~lim (n + 1),,+i r/n (3) matics are considered; exhaustiveness is not the goal. Nay, the breadth and depth of our treatment of e has been cho- discovered by Brothers and Knox in 1998 [11]. Figure 1 dis- sen to convey the versatility of this remarkable number and plays the sequences involved in equations (1) and (3). The to whet the appetite of the reader for further investigation. superior convergence of equation (3) is apparent. In light of the fact that both equations (1) and (3) pro- The Cast vide rational approximations to e, it is interesting to note Unlike its elder sibling ~r, e cannot be traced back through that 87~ = 2.71826 provides the best rational approxirna- 323 the mists of time to some prehistoric era [8]. Rather, e burst tion to e, with a numerator and denominator of fewer than into existence in the early seventeenth century in the con- four decimal digits [12]. (Note the palindromes!) Consider- text of commercial transactions involving compound inter- ing that the fundamental constants of nature (speed of light est [5]. Unnamed usurers observed that the profit from in- in vacuo, mass of the electron, Planck's constant, and so terest increased with increasing frequency of compounding, on) are known reliably to only six decimal digits, this is re- but with diminishing returns. markable accuracy indeed. Mystically, if we simply delete

10 THE MATHEMATICAL INTELLIGENCER 2006 Springer Science+Business Media, Inc. 1 i i i i i i 6 o 0 0 0 0 2.7 l e

2.6

2.5

2.4 = Eq. 1 o= Eq. 3 2.3

2.2

2.1

2 i i I i I 1 I I I 2 4 6 8 10 12 14 16 18 20 n Figure I. Exponential sequences.

the last digit of both numerator and denominator then we ~, 2k+2 t~ s732, the best rauonal" approximation to e using fewer e = (5) k=0 (2k)! han three digits [13]. Is this to be regarded as a singular property of e or of base 10 numeration? Figure 2 displays the partial sums of equations (4) and (5) In 1669, Newton published the famous series represen- and clearly reveals the enhanced rate of convergence. A va- tation for e [14], riety of series-based approximations to e are offered in [15]. Euler discovered a number of representations of e by con- e---= 2 ~t 1 = tinued fractions. There is the simple continued fraction [16] k 0 1 1 1 1 1 1+1+ ~.(1+ ~-.(1+ ~-.(1+ ~-.(1+ .'')))), (4) e = 2 + 1 , (6) 1+ 2+ 11 established by application of the binomial expansion to 1+ 1 1+ 4+ l+.__.k~_~1 equation (1). Many more rapidly convergent series repre- 1+ r~... sentations have been devised by Brothers [14] such as or the more visually alluring [5] BRIAN J. McCARTIN studied Applied Mathe- matics at the University of Rhode Island and e = 2 + 1 (7) went on to a Ph.D. at the Courant Institute, 1 + 2+-- 2 8 3+ New York University. He was Senior Research 4+ 4 5+6+ 0 6 7 Mathematician for United Technologies Re- 7+8+... search Center and Chair of Computer Science at RPI/Hartford before joining Kettenng Univer- In 1655, John Wallis published the exhilarating infinite sity in 1993. He received the Distinguished product Teaching Award from the Michigan Section of rr 2244668810101212141416 the MAA in 2004. Along the way, he also ...... (8) 2 133557799111113131515 eamed a degree in music from the University of Hartford (1994). To fill in the details see However, the world had to wait until 1980 for the "Pip- http://www.ketted ng,edu/- bmccarti. penger product" [17] Applied Mathematics e Kettering University 2 Flint, MI,' 48504-4808 USA \l J\ 3! \5 5 7 7/\ 9 9 11 11 13 13 15 ~/ '''. (9)

9 2006Springer Science+Business Media, Inc, Volume 28, Number 2, 2006 1 1 2.8 i i i i i i O O Q o Q t o 2.6 e

2.4

2.2

= Eq. (4) o = Eq. (5) 1.8

1.6

1.4

1.2

1 I f I I I I 0 1 2 3 4 5 6 n Figure 2. Exponential series.

In spite of their beauty, equations (8) and (9) converge Barel-e Transcendental very slowly. A product representation for e which converges As noted above, e is transcendental, but just barely so [21, at the same rate as equation (4) is given by [18] 22]. In 1844, Liouville proved that the degree to which an algebraic irrational number can be approximated by ratio- U 1 ----- 1; Un+ 1 = (T/ 4- 1)(Un -I- 1) nals is limited [23], so that, from the point of view of ratio- [I un+ 1 2 5 16 65 326 nal approximation, the simplest numbers are the worst [24]. e ...... (10) n=l U*I 1 4 15 64 325 Liouville used this property--that if an irrational number is rapidly approximated by rationals then it must be transcen- An Interesting Inequalit-e dental-to construct the first known transcendental number, In 1744, Euler showed that e is irrational by considering the the so-called Liouville number L = 10 -1! + 10 -2! + 10 -3! 4- simple infinite continued fraction (6) [19]. In 1840, .. 10-m! + -. In 1955, Klaus Roth provided the defini- Liouville showed that e was not a quadratic irrational. Finally, tive refinement of Liouville's theorem by finding the ulti- in 1873, Hermite showed that e is in fact transcendental. mate limit to which algebraic irrationals may be approxi- Since then, Gelfond has shown that e ~ is also transcen- mated by rationals [19]. For this work, he was awarded the dental. Although now known as Gelfond's constant, this Fields Medal in 1958. number had previously attracted the attention of the influ- To phrase these results quantitatively, for any real num- ential nineteenth-century American mathematician Benjamin ber 0 < s~ < 1 and any fraction p/q in lowest terms, let R Peirce, who was wont to write on the blackboard the fol- denote the set of all positive real numbers r for which the lowing alteration of Euler's identity [2, 5]: inequality

'= ~e ~, (11) 1 0 < I~ -- P l < qr (14) then turn to the class and cryptically remark, "Gentlemen, we have not the slightest idea what this equation means, but we possesses at most finitely many solutions. Then, define the may be sure that it means something very important" [4]. Liouville-Roth constant (irrationality measure) as But what of ~r~? Well, it is not even known whether it is rational! Niven [20] playfully posed the question "Which is r(~ ~- inf r, (15) larger, e ~r or rre?''. Not only did he provide the answer r~R e ~ > ITe, (12) that is, the critical rate threshold above which {: is not ap- proximable by rationals. he also established the more general inequalities Then, the above may be summarized as follows [22]: /3>a->e~ c~/3>/3~; e->/3>a>0~/3~>~/3, (13) {i: when {: is rati~ where e plays a pivotal role. This result is displayed graph- r(~ = when {: is algebraic of degree > 1, (16) ically in Figure 3. 2 when s~ is transcendental.

12 THE MATHEMATICALINTELLIGENCER xY-yX=O

10 I i I I I I I 9=i "/ >0

I 6

>, 5

4 <0 <0

3 (e,e)

0 i I I I I I I I 0 1 2 3 4 5 6 7 8 10 x Figure 3. e" > ~'~.

2.8 I I I i I

,e..- e

2.6

2.41

2.2

1.8

1.6

1.4 = i i f i 0 1000 2000 3000 4000 5000 6000 n Figure 4. Primorial limit.

2006 Springer Science+ Business Media, Inc.. Volume 28, Number 2, 2006 1 3 More germane to our present deliberations, it is known that [211 r(e) = 2. (17)

Thus, e dwells in the twilight zone of overlap between the algebraic irrationals and the transcendental numbers.

The Primor-e-al Letting Pn denote the nth prime number, we define the primorial of Pn as follows [25]: n B P n*~ ~ H Pk. (18) k--1 Figure 6. Spiral compass. Thus, the primorial sequence proceeds as follows: 2, 6, 30, 210, . . . An integer is called a primorial prime if it is ing from x = 1, ~r is defined as the ratio of two lengths (cir- a prime of the form p,,#-+ 1. For example, 211 = cumference to diameter of a circle). However, with refer- 2 3 5 7 + 1 is a primorial prime. It is not known whether ence to Figure 5, e may also be defined as the ratio of two there are infinitely many primes p for which p~ -+ 1 is prime lengths [28]: or whether there are infinitely many primes p for which p~ + 1 is composite. A0=tana~ r2 =e. (22) However, Ruiz [26] has established the following re- rl markable limit relating e to the primes: (For example, if a = rr/4, then A0 should be chosen to be lim (p,~)l/p, = e. (19) one radian.) But since, according to equation (21), the spi- n---+ac ral requires e as part of its very definition, is this not a cir- As Figure 4 shows, the approach to the limit is not nearly cular argument? as tame or as speedy as that in Figures 1 and 2. No, it is a spiral argument, for an equiangular spiral A related result certainly worthy of note is the following: may be drawn without reference to equation (21) [7]! Fig- Let An and Gn denote the arithmetic and geometric means, ure 6 displays a spiral compass designed for this purpose. respectively, of the integers 1, 2, 3, . . . ,n. Then [18] The compass point is located at O and is free to slide A n e smoothly in the slotted rod BC The wheel F, which lies lim - (20) n--,~ Gn 2' in a plane perpendicular to the page, may be locked in place at the desired angle relative to this rod. A small han- e-quiangular Spiral dle protrudes at G which permits grasping the apparatus Figure 5 displays the equiangular (logarithmic) spiral de- between the thumb and two first fingers. Its sharp edge fined by keeps the wheel from slipping sideways. Hence, as the wheel revolves it maintains a constant angle with the slot- r = e c~ ~-e (21) ted rod and thereby traces out an equiangular spiral of and possessing the distinguishing characteristic of intersect- any desired eccentricity. ing any radial line drawn from its center at a constant an- The equiangular spiral arises in many natural settings (and gle ~. This spiral has many remarkable self-reproducing some unnatural ones as well: the beginning of the Yellow properties, such as that of being its own evolute [5]. Jakob Brick Road is reputed to be a golden spiral). One of these is Bernoulli (brother and sometimes-adversary of Euler's shown in Figure 7. The cauliflower (Broccoli Romanesco) teacher, Johann [27]) was so taken by it that he had it placed there displayed is resplendent with equiangular spirals. Deli- on his gravestone with the inscription "Eadum mutata ciously enough, it is also a rich source of vitamin E! resurgo (Though changed, I shall arise the same)." While e has the straightforward geometrical interpretation Stirl-e-ng's Formula as delimiting a unit area under the hyperbola y = 1/x start- In 1730, Stirling published the asymptotic formula [29]

n!- e ,z. n n . 2~n. (23)

In addition to providing an intriguing connection between ~* 0t ~r and e, it also affords the limit n e = lim (24) n~ (n!) 1/~z'

However, it should be pointed out that equation (24) can be obtained directly from equation (1) by elementary means and, furthermore, that equation (20) is now an immediate Figure 5. Spira mirabilis. consequence of equation (24). Formula (23) bears so many

14 THE MATHEMATICAL INTELLIGENCER In that case,

f'(x + y) = f'(x)f(y) = f(x)f'(y), (26) which upon rearrangement becomes

f'(x) c .f'@ ~ f(x) = Ae cx, (27) f(x) f(y) Since equation (25) clearly implies that f(0) = [./'(0)] 2, we must have either f(0) = 0 = A off(0) = 1 = A, thereby pro- ducing the two solutions as required. Observe that, from equation (27), the emergence of the exponential solution was a direct consequence of its being the eigenfunction of the differentiation operator:

d (D- c)y(x)=0 ~y(x)=AeCX; D=-- dx" (28)

This same fundamental property provides the exponential function with a central role throughout the vast field of differential equations and their applications. For example, consider the first-order, constant coefficient, scalar equation [32]

y'(x) - ry(x) =f(x). (29) Multiplication by the integrating factor/, ~ e-ru and invo- cation of the eigenfunction property produces the self- adjoint form Figure 7. Vitamin e. [e-rc'iy(x)] ' : e-rX f(x), (30) remarkable relationships to the foregoing as to make it pos- which upon integration leads directly to the general solution itively eclectic. To begin with, Wallis's formula (8) was used y(x) = e rx dx, (31) by Stifling to determine the constant factor in his asymptotic fe-rx f(x) formula. Furthermore, Pippenger used Stirling's formula in where we have utilized the shorthand f4Xx)dx ==- [x4x.~)d~. deriving his product (9). Moreover, Stirling's formula is an Let us recast this result in operator form. Equation (29) essential ingredient of Borwein's derivation of equation (17) becomes for the Liouville-Roth constant for e. Lastly, both Ruiz's eval- uation of the primorial limit (19) and the determination of (D - r)y(x) = f(x), (32) the arithmetic/geometric mean limit (20) hinge upon the in- while its solution, equation (31), becomes vocation of Stirling's formula. Let us try to gain an intuitive feel for the accuracy of Stir- 1 9S(x) = e rx re- rx [(x) d& (33) ling's formula for moderate values of n. Specifically, 100! = y(x) = O~- r 9.3326 9 9 9 X ~57, 10 while Stirling's formula produces 100! providing an explicit representation of the inverse of the dif- 9.3248 9 9 - •10 1",7- with an error of less than --Ye.1 0 This also 10 ferential operator D- r. allows us to appreciate the immensity of n! for even We may exploit this observation in order to bootstrap to moderate values of n. For example, the age of the Universe a solution procedure for second-order equations with con- is believed to be O(1017) seconds, and the number of atoms stant coefficients. Consider, for example, in the visible Universe has been estimated to be O(1079). Yet, we must temper our awe of the magnitude of n! with y"(x) - 3y'(x) + 2y(x) = xe ~':, (34) the realization that the number of possible chess games has or, in operator fom~, been estimated to be O(101~176 [30]. (D 2 - 3D + 2)y(x) -- xe x. (35) eigenfunction Factorization followed by inversion then leads to The functional equation 1 1 ./`(x + j,) ---f(x)/`(y) (25) y(x) .... xe < (36) D-1 D-2 is known as the exponential equation of Cauchy [31]. If.f(x) An initial application of the inversion formula (33) pro- is bounded on a set of positive measure then the only so- duces the intermediate result lutions of (25) are f(x) = 0 and f(x) = e% where c is an arbitrary constant. This is easy to establish if we make the --1 " xe x= e2 x f e 2XxeX dx= -(1 +x)eX; (37) more restrictive assumption that f(x) is differentiable. D-2

@ 2006 Springer Science+Business Media, Inc., Volume 28, Number 2, 2006 15 then a second application provides a particular solution to exponential Generating Functions equation (34): The sequence {an}n=0 is said to be generated by the function g(x) if they are related by [35] 1 y(x) = -- [-(1 + x)e x] D-1 g(x) = ~ anX n (47) n=0 = -e x f e-X(1 + x)e x dx = -1(1 + x)Ze x. (38) For example, (1 + x) m generates the binomial coefficients (m). Clearly, this symbolic process generalizes to nth-order constant coefficient operators. Likewise, g(x) is called the exponential generating func- However, it is frequently more natural to recast a higher- tion for the sequence {an}n= 0 if order equation as a system of first-order differential equations [33]: g(x) = -- x n (48) n o n! 2'(t) = A2(t) + f(t); 2(0) = 20. (39) For example, the Bernoulli function B(x) is the exponential Defining the matrix exponential as generating function for the Bernoulli numbers Bn:

e tA ~ -- A n, (40) x _ Bn" --. (49) B(x)- e x _ 1 n! n=O n! n=O2 we arrive at a solution analogous to equation (31), The Bernoulli numbers arise in a dizzying variety of an- alytical and combinatorial contexts [9], like Faulhaber's for- 2(t) = etA2o + e(t-')A f(r) dr. (41) mula for the sums of powers of integers:

kP=P~ 1 (--1)P-k+l BP-k+I P' n k. (50) This useful result is called the variation of parameters for- k= 1 k= 1 la! (p- k + 1)! mula. Part of its allure is that it expresses the solution in a form whereby the influence of the initial conditions, 20, and In a later section, I will explore the use of exponential the external influences, f(t), have been isolated and hence generating functions in combinatorics and graph theory. But may be studied separately. first let us explore an important occurrence of the Bernoulli function in its own right [36]. From Continuous to Discr-e-te The preceding section detailed an operational approach to Bernoull-e Function and Singular Perturbations differential equations (continuous models) founded on the Many natural phenomena are governed by differential equa- eigenfunction property of the exponential function. An anal- tions whose highest derivative is multiplied by a small pa- ogous symbolic calculus may be developed for the study of rameter E. These include [37] pollutant dispersal in a river difference equations (discrete models). Appropriately enough, estuary, vorticity transport in incompressible flow, atmo- the bridge between these parallel universes (the continuous spheric pollution, kinetic theory of gases, semiconductor de- and the discrete) is afforded by e [34]. vices, groundwater transport, financial modelling, melting First, define the shift operator E, phenomena, fluid flow over an airplane, and turbulence transport. kf(x) =- f(x + b), (42) This field of applied mathematics is called singular per- and the forward difference operator A, turbation theory because the correct physical solution for Af(x) =- f(x + h) - f(x). (43) small E bears little resemblance to the solution obtained by simply setting E to zero. A prototypical problem is provided Then by Taylor's theorem, E is expressible in terms of D of by [381 the previous section as e. u'(x) + u(x) = 0 ~ ui = e -Ax/E Ui-1, (51) 1 + A = E = e hD. (44) where the subscript notation indicates that the problem has Equation (44) may now be used to transform back and been discretized on a mesh of width Ax. The exact solu- forth between the continuous world of D and the discrete tion afforded by equation (51) is shown in Figure 8 with world of A: u0 = u(0) = 1. The steep front that is on prominent display 1 there is called an initial layer. Df(x) = -~ln(1 + A)f(x) If we approximate the differential equation by the ex- 1 A2 A3 plicit (forward) Euler scheme [39], we obtain = --[A- -- + ..... ]f(x), (45) h 2 3 bl i -- Ui_ 1 I~" -]- bli- 1 = 0 ~ U i = (1 - Ax/e) ui-1 (52) h 2 Ax Af(x)=(e bD- 1)f(x)=[hD+ --D 2+ ...]f(x). (46) 2I Comparison of the solutions in equations (51) and (52) re- Corresponding relations may be derived involving both veals that this is equivalent to the use of the first-order Tay- backward and central difference operators, but the above lor series approximation to the exponential function, e z -~ should be sufficient to convey the essential role of e as in- 1 + z. This approximation is shown in Figure 8 for Axle = termediary between these two distant lands. 1.9608 and is clearly seen to be inadequate.

16 THE MATHEMATICALINTELLIGENCER If we instead insert a strategically placed Bernoulli func- enclosing an angle of approximately 40 ~ [42]. Outside the tion, B(-Ax/E), into our approximation wake, the effect of the duck's motion is "exponentially small"; inside and on the caustic, its effect may be obtained ~. B(-Ax/~). ui- ui-1 + ui-1 through asymptotic approximations [43]. However, as first kx observed by Stokes in 1847, these exponentially small terms =0 ~ ui= e -axlE" ui-1, (53) must be accounted for in order to produce refined asymp- we obtain the exact solution! This spectacularly successful totic approximations. trick is known as exponential fitting [40]. Quantitative accuracy is not the only concern of such so- Let us now employ the implicit (backward) Euler scheme called exponential asymptotics. The very existence of a so- to provide our approximation to the differential equation lution may hinge on properly accounting for exponentially small terms. For example, in crystal growth the formation ui - ui- ~ 1 e. + ui=0 ~ ui= "ui-1. (54) of dendritic fingers is governed by an exponentially small Ax 1 + Axle term in the surface energy between solid and liquid [43]. Comparison of the solutions in equations (51) and (54) This is an area of intense current interest in applied math- now reveals that this is equivalent to the use of the ematics [42]. (0,1)-Pad6 approximation to the exponential function [41], e z ~ 1/(1 - z). Revisiting Figure 8, we find that, while the exponential Transform implicit scheme vastly improves over the explicit scheme, it Let us now return to the subject of exponential generating is still inadequate for many applications. functions. In combinatorial analysis [44], generating func- Introduction of a strategically placed Bernoulli function, tions are used for problems involving combinations while B(Ax/E), into our approximation exponential generating functions find application in problems involving permutations. However, exponential gener- E" B(Aoc/ff.) IXi -- IXi-1 + U i ~-- 0 ~ U i = e -• ui-i (55) kx ating functions find wider application to graphical enumeration problems [45]. again produces the exact solution. The crucial role played Let A(x) = ~=1 anx"/n! be the exponential generating by the Bernoulli function in the approximation of singularly function of the sequence {au}~=l and B(x) = "Z~=l bnxn/n! perturbed differential equations is thereby revealed. be the exponential generating function of the sequence {b,,}~=l. If 1 + B(x) = e a(x) then {bn}n=l is called the expo- exponential Asymptotics nential transform of {an}~=l and {an}n=1 is called the loga- Continuing with the theme of the previous section--just be- rithmic transform of {bn}~=l [46]. cause a term is small does not imply that it can be neglected Riddell's formula relates the number of graphs with n ver- in a respectable mathematical analysis of a physical prob- tices to the number of such graphs which are connected [47]. lem. Consider Figure 9, where we observe a duck moving It is stated in [48] that "if a, is the number of connected la- along the surface of a pond, leaving in its wake a "caustic" beled graphs with a certain property, then bn is the total num-

g i i i

0.8

0.6

0.4

0.2

-0.2 = exact solution o = exponential fitting = explicit Euler -0.4 = implicit Euler

-0.6

-0.8

-1 I I I I I I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Figure 8. Another Bernoulli miracle!

2006 Springer Science+Business Media, Inc., Volume 28, Number 2, 2006 17 Figure 9. Duck-e with attendant wake. ber of labeled graphs with that property." Letting the prop- and define at = ln/zt (k = 1, . . . n). erty in question be "having exactly two connected compo- 3. Solve by least squares for q, . . . ,c, in the N linear nents," we see that as stated this is clearly false. (a,, = 0 for equations all n so that Riddell's formula implies that b, = 0 for all n, but C 1 q- C 2 -F- " " " q- C n =j~ b2 = 1.) Unfortunately, this error is repeated by Weisstein [46]. Sloane and Plouffe cite Harary and Palmer [45] who in turn /d'l C1 q- /J'2C2 q'- " " " q- [tnCn = "/q (59) i cite Riddell [49]. However, inspection of Riddell's original 1-1'1\~1 El Jr- 1"1"~"-1 C2 "}-''" "Jr- [d'~Y-1 Cn =/N-I- work [49] reveals that he never applied the exponential transform to any property other than connectedness. The correct Depending on the size of n, we may either solve equation generalization of Riddell's formula requires amendment of the (58) by directly applying a polynomial root-finding algo- above: "then bn is the total number of labeled graphs with rithm or by recasting it as an eigenvalue problem for the n vertices whose connected components possess the same companion matrix [50] property." Thus, if one restricts oneself to properties inher- ited by the connected components of a graph, then Riddell's --/31 1 0 9 9 9 0 formula may be confidently used as an enumerative tool. For --/32 0 1 9 9 9 0 (60) example, if we let the property be "even-ness" then Riddell's -/33 0 0 9 9 9 . : : ".. formula correctly relates the number of Eulerian graphs to - ,, 0 0 ''' the number of even graphs (see [47] for the definitions of these graph-theoretic concepts). Otherwise, employment of Modern-day applications of Prony's method include dig- Riddell's formula produces erroneous results. ital-filter design, radar and sonar signal processing, determination of atmospheric transfer functions, and even radia- Pron-e's Method tion therapy planning for cancer patients [51]. The original It is sometimes necessary to approximate a function by a algorithm of Baron de Prony [52] dates back to 1795 and sum of exponential functions. Essentially a nonlinear prob- was concerned only with the case N= 2n. Unlike his con- lem, this may be accomplished by transforming it to the temporaries Monge and Fourier, de Prony refused to join problem of finding the zeros of a certain polynomial. This Napoleon's invasion of Egypt. Only his wife's close friend- technique is known as Prony's method [39]. ship with Josephine averted the anticipated dire conse- Given f(x) at x= 0,1, . . . , N- 1, we seek an expo- quences of such lack of military zeal [53]. nential approximation Probabl-e f(x) .~ Cl ealx + .'. + one a,,x, (56) e has the curious habit of asserting itself in strange proba- where N --> 2n. Defining~ =f(j), Irk := e% Prony's method bilistic contexts [54]. For example, suppose that numbers are proceeds as follows: selected at random from the interval [0,1]. What is the expected number of draws necessary for the sum of these 1. Solve (by least squares if necessary) for ill, 9 . . fi,, in numbers to exceed 1? There are several elementary proofs the N- n linear equations [54, 55, 56] that the answer is emphatically e. .fiz + fn-1/31 + " "" +.5/3,z = 0 Then there is the old chestnut of the "secretary problem" ! (57) [22, 54]. Here, it is required to select a new secretary from dr1 +dr 2/31 + "'" +dr,, 1/3, = 0. a pool of n applicants. Candidates are interviewed sequen- tially and the decision whether or not to hire an individual 2. Find the roots/~l, 9 9 /.t,, of must be made at the conclusion of the interview. If the can- /jn -l- /31///l 1 q_ . . . _}_ /3,z 1/-t q- /3,t ~--- O, (58) didate is not selected then he/she is excluded from further

18 THE MATHEMATICAL INTELLIGENCER consideration. If this process ever reaches the last applicant to model the number of arrivals for a queue in a given in- then that individual is automatically hired. The goal is now terval of time and, correspondingly, the exponential distri- to devise a strategy that maximizes the probability of hiring bution is used to model the actual arrival times. Thus, they the most qualified applicant. There turns out to be a sim- give rise to mathematical models of supermarket waiting ple optimal strategy: for some integer k (to be determined lines and car accidents. If the independent variable is not as part of the optimization problem) with 1 -< k < n, inter- time but space, then the exponential distribution can be view and reject the first k applicants. Then, from the re- used to model distance between roadkill or between muta- maining n- k applicants, choose the first one that is the tions on a strand of DNA. best seen to date. In the limiting case of large n, the opti- The exponential distribution possesses a special property mal value is k ~ n/e ~ 0.368n and the probability of find- that endows it with great utility in reliability theory [60, pp. ing the best applicant is asymptotically 1/e (about 36.8%). 82-83]. This can be seen by computing the probability of This is but one of a variety of optimal stopping problems failure of some physical device during the interval (x, x + dx), [22]; remarkably, this very result has been applied to the al- assuming that it did not fail prior to time t: location of cadaveric kidneys tor transplantation [57]. ,/'(xlX >- t) dx=f(x- t) dx; x >- t. (63) Well, our newly hired secretary is working out fine until the Christmas luncheon, at which one too many spiked eggnogs where X is the random failure time of the device. is imbibed. Upon returning to the office, we are confronted Thus, the conditional failure rate, [3(0, defined as with the "drunken secretary problem." Waiting on the desk are [3(t) =- f(t]X >- t) =f(0) = c, (64) n different letters, each with a corresponding addressed envelope. What is the probability that he/she will produce a "de- is independent of t. This is called the memoryless property, rangement" whereby no letter is placed in its correct envelope? and, significantly, only the exponential distribution possesses Euler first posed this problem, answered it, and showed that it. This implies that, in any application where the failure rate the required probability asymptotically approaches 1/e [21. is approximately constant, the exponential distribution must Of course, the flip side of this problem is: What is the arise. This is evidenced by its frequent appearance in mod- probability, p, that a permutation of n distinct objects has eling the "middle years" of the lifetime of diverse systems at least one fixed point? According to the above, p ~ 1 - ranging from industrial machines to human beings. 1/e ~ 0.632. We have hardly exhausted all of the appear- The exponential function is also used to define the mo- ances of e in probability [54], but it is time we moved on ment generating t\mction of the statistical distribution with to her sister discipline of statistics. density function f(x) [60, p. 116]:

(s) =- f(x) e -~x dx ~ m,, ~ E{X ~'} = ~~ (65) Statisticall-e x. While occurrences of e throughout probability theory might where E is the expected value. Consequently, qb(s) is the be characterized as sporadic, its influence on statistical the- exponential generating function of the moment sequence ory has been pervasive. First introduced by DeMoivre in In point of fact, generating functions were first 1733 as the limit of the binomial distribution [58], the nor- {mn},~=l. >/ ) exploited by Lagrange and Laplace in their probabilistic real distribution, N(x;tx,~)=(o-~7~) le-CV-**>'{e~'), has investigations [59]. Providing yet another unifying thread, come to dominate the statistical landscape. Spurred on by I note that Stirling's formula is an essential ingredient in Gauss's 1809 theory of errors of observation and Laplace's one of the proofs of the DeMoivre theorem alluded to publication in 1812 of the Central Limit Theorem [59], study above [60]! of the normal distribution became virtually synonymous with statistical inquiry. This fixation on the normal distribution reached an abnormal pitch in the late nineteenth century at Family Tr-e the hands of Adolphe Quetelet and Sir Francis Galton [59]. Over the course of four centuries, e has dutifully obeyed the This frenzy was tempered somewhat in the twentieth cen- biblical directive "be fruitful and multiply" and spawned an tury by Galton's prot~g~ Karl Pearson and his nemesis Sir illustrious lineage. Among its distinguished progeny [35, 61]: Ronald Fisher, but to this day the normal distribution holds hyperbolic functions: sinh(z) = (e z- e-Z)/2, etc. a special place in the hearts and minds of statisticians. trigonometric functions: cos(z) = (e'Z+ e-'e)/2, etc. e is intimately involved with at least two other statistical Bernoulli function: B(x) = x/(e x- 1) distributions [60], one discrete, the Poisson distribution with Gaussian function: G(x; c~,[3)= e [~x-~//31-~ parameter a (mean) 2 error function: eft(x) = ~7 f~ e-Fdt a h Prob{X= k} = e -a" -- (61) Gamma function: F(z) = J'~ e-tt~-ldt (Re z> 0) le!' exponential integral: El(x) = f**~,: 5dt and the other continuous, the exponential distribution with orthogonal polynomials: parameter c (1/mean) Hermite: weight w(x) = e -'-' on (-%0c) Laguerre: weight w(x) = x~e x on (0 ac), c~ > -1 f(x) = ce ~:~. U(x), (62) integral transforms: where U(x) denotes the Heaviside unit step function. Fourier: f~:~ e*a:r.f(y)dy These two distributions are related by the theory of sto- Laplace: f~: e ':r f(y)dy chastic processes [60], where the Poisson distribution is used Gauss: f_~:~ e ~" y~e f(_v)dy

@ 2006 Springer Science+Business Media, Inc., VoLume 28, Number 2, 2006 19 Figure I0. Google-plexed.

It should be clear from the above that one would be challenged to find a branch of pure or applied mathematics which has not felt the enriching presence of e. Readers are invited to add their own favorite branches to this al- ready robust family tree.

Conclusion The time has come to end our stroll through the Garden of e-den, even though we have yet to taste many of its most Figure II. e-gotism. succulent fruits, such as exponential splines [62]. However, I would like to conclude my p-e-an to e in a lighter vein. REFERENCES The initial decimal digits of e are quite easy to remem- [1] C. C. Gillispie, Biographical Dictionary of Mathematics, Charles ber: 2.7 1828 1828 45 90 45. In spite of this, many mnemon- Scribner's Sons, New York, 1991. ics based on the number of characters of each word have [2] W. Dunham, Eder: The Master of Us All, Mathematical Associa- been developed. A particular favorite is: "It enables a num- tion of America, Washington, DC, 1999. skull to memorize a quantity of numerals" [63]. For the more [3] U. G. Mitchell and M. Strain, The Number e, Osiris 1 (1936), sophisticated, I offer the pleasingly self-referentiah "I'm form- 476-496. ing a mnemonic to remember a function in analysis" [13]. [4] J. L. Coolidge, The Number e, American Mathematical Monthly Of course, anything can be taken to excess, as is evidenced 57 (1950), 591-602. by the gargantuan 40-digit mnemonic of [64]. [5] E. Maor, e: The Story of a Number, Princeton University Press, Internet giant Google holds the distinction of having Princeton, NJ, 1994. transformed a mathematical noun (the googol [65]) into a [6] A. H. Bell, The Exponential and Hyperbolic Functions and TheirAp- verb. As further evidence of their mathematical pedigree, plications, Pitman & Sons, London, 1932. when filing the registration for their 2004 initial public of- [7] M. E. J. Gheury de Bray, Exponentials Made Easy, Macmillan & fering of stock (IPO) with the Securities and Exchange Com- Co., London, 1921. mission (SEC), they chose not to state the number of shares [8] P. Beckmann, A History of ~-, Barnes & Noble, New York, 1993. to be offered. Instead, they declared $2,718,281,828 as their [9] J. H. Conway and R. H. Guy, The Book of Numbers, Springer- estimate of the money that would be so raised. Verlag, New York, 1996. Later that same year, from deep within their corporate [10] M. Brede, On the Convergence of the Sequence Defining Euler's headquarters in Mountain View, Califomia (the Googleplex), Number, Mathematical Intelligencer 27 (2005), 6-7. a most curious recruitment strategy was hatched. Billboards [11] H. J. Brothers and J. A. Knox, New Closed-Form Approximations and banners, such as that in Figure 10, appeared anonymously to the Logarithmic Constant e, Mathematical Intelligencer 20 in Silicon Valley and Cambridge, Massachusetts. After deci- phering the puzzle (whose solution begins strangely enough (1998), 25-29. at the 101st digit of e) and visiting 7427466391.com, one was [12] D. Wells, The Penguin Dictionary of Curious and Interesting Num- eventually led to a solicitation of employment for Google. bers, Penguin Books Ltd., Middlesex, England, 1986. Even though e finally has its own biography [5], it has [13] M. Gardner, The Unexpected Hanging and Other Mathematical Di- traditionally had to dwell in the shadow cast by the head- versions, Simon and Schuster, New York, 1969. line-grabbing ~r. Witness the fact that in [12], 7r has 8 pages [14] H. J. Brothers, Improving the Convergence of Newton's Series devoted to it, while e is allocated a paltry 1/2 page! How- Approximation for e, College Mathematics Journal 35(1) (2004), ever, the situation has improved to the point where e now 34-39. has its own Web page [66], from which Figure 11 is bor- [15] J. A. Knox and H. J. Brothers, Novel Series-Based Approxima- rowed. tions to e, College Mathematics Journal 30(4) (1999), 269-275. [16] C. D. Olds, The Simple Continued Fraction Expression for e, Amer- ACKNOWLEDGMENTS ican Mathematical Monthly 77 (1970), 968-974. The author thanks Mrs. Barbara McCartin for her dedicated [17] N. Pippenger, An Infinite Product for e, American Mathematical assistance in the production of this paper. Figure 7 is cour- Monthly 87(5) (1980), 391. tesy of http://static.filter.com while Figure 9 is courtesy of [18] X. Gourdon and P. Sebah, The Constant e and Its Computation, http://FreeNaturePictures.com. http://numbers.computation.free.fr/Constants/E/e.html.

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Robbins, What Is Mathematics?, Oxford Uni- [49] R. J. Riddell, Jr., Contributions to the Theory of Condensation, versity Press, New York, 1941. Ph.D. Dissertation, University of Michigan, UMI Dissertation Ser- [24] G. H. Hardy and E. M. Wright, An Introduction to the Theory of vices, Ann Arbor, MI, 1951. Numbers, Third Edition, Oxford University Press, London, 1954. [50] A. S. Householder, The Numerical Treatment of a Single Nonlin- [25] P. Ribenboim, The New Book of Prime Number Records, Springer, ear Equation, McGraw-Hill, New York, 1970. New York, 1996. [51] K. HolmstrOm, A. AhnesjO, and J. Petersson, Algorithms for Ex- [26] S. M. Ruiz, A Result on Prime Numbers, Mathematical Gazette 81 ponential Sum Fitting in Radiotherapy Planning, Technical Report (1997), 269-270. IMa-TOM-2001-02, Department of Mathematics and Physics, [27] E.T. Bell, Men of Mathematics, Simon and Schuster, New York, 1965. M~larden University, Sweden (2001). [28] H. V. Baravalle, The Number e--The Base of the Natural Loga- [52] G. R. de Prony, Essai experimental et analytique, Joumal de rithms, The Mathematics Teacher 38 (1945), 350-355. I'E-cole Polytechnique, 1(22) (1795), 24-76. [29] B. Friedman, Lectures on Applications-Oriented Mathematics, [53] J. J. O'Connor and E. F. Robertson, Gaspard Clair Frangois Holden-Day, San Francisco, 1969. Marie Riche de Prony, http://www.groups.dcs.st-and.ac.uk/history/ [30] G. H. Hardy, Ramanujan, AMS Chelsea, Providence, RI, 1999. Mathematicians/De_Prony.htmL [31] J. Aczel, On Applications and Theory of Functional Equations, [54] H. S. Schultz and B. Leonard, Unexpected Occurrences of the Academic Press, New York, 1969. Number e, Mathematics Magazine 62(4) (1989), 269-271. [32] H. Bateman, Differential Equations, Chelsea, New York, 1966. [55] N. MacKinnon, Another Surprising Appearance of e, Mathemati- [33] R. E. O'Malley, Thinking About Ordinary Differential Equations, cal Gazette 74(468) (1990), 167-169. 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Barel, A Mnemonic for e, Mathematics Magazine 68 (1995), 253. [42] D. Mortimer, Research on Asymptotics, http://www.maths.ox.ac. [65] E. Kasner and J. Newman, Mathematics and the Imagination, Si- uW- mortimer/research/davidresearch.shtml. mon and Schuster, New York, 1967. [43] A. C. Fowler, Mathematical Models in the Applied Sciences, Cam- [66] D. Sapp, The e Home Page, http://www.mu.org/-doug/exp/ bridge University Press, Cambridge, 1997. etopl0.html.

2006 Springer Science+Business Media, Inc., Volume 28, Number 2, 2006 ~)1