E (mathematical constant) explained

Euler's number
Type:	Transcendental
Approximation:	2.71828...
Discovery Date:	1685
Discovery Person:	Jacob Bernoulli
Discovery Work:	Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685

The number is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithm function. It is the limit of

( 1+1 /n) ^{n} as tends to infinity, an expression that arises in the computation of compound interest. It is the value at of the (natural) exponential function, commonly denoted e ^{x.} It is also the sum of the infinite series$$e\; =\; \backslash sum\backslash limits\_^\; \backslash frac\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots.$$There are various other characterizations; see and .

The number is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, though this can invite confusion with Euler numbers, or with Euler's constant, a different constant typically denoted

\gamma . Alternatively, can be called Napier's constant after John Napier.^[1] The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest.^[2] e ^{i\pi} +1=0 and play important and recurring roles across mathematics.^{[4] [5]} Like the constant, is irrational, meaning that it cannot be represented as a ratio of integers, and moreover it is transcendental, meaning that it is not a root of any non-zero polynomial with rational coefficients. To 30 decimal places, the value of is:

Definitions

The number is the limit $$\backslash lim\_\backslash left(1+\backslash frac\; 1n\backslash right)^n,$$an expression that arises in the computation of compound interest.

It is the sum of the infinite series$$e\; =\; \backslash sum\backslash limits\_^\; \backslash frac\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots.$$

It is the unique positive number such that the graph of the function has a slope of 1 at .

One has $$e=\backslash exp(1),$$ where

\exp is the (natural) exponential function, the unique function that equals its own derivative and satisfies the equation \exp( 0 ) =1 . Since the exponential function is commonly denoted as x\mapsto e ^{x,} one has also$$e=e^1.$$

The logarithm of base can be defined as the inverse function of the function

x\mapsto b ^{x.} Since b = b ^{1,} one has log _{b} b = 1 . The equation e = e ^{1} implies therefore that is the base of the natural logarithm.

The number can also be characterized in terms of an integral:$$\backslash int\_1^e\; \backslash frac\; x\; =1.$$

For other characterizations, see .

History

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base

e . It is assumed that the table was written by William Oughtred. In 1661, Christiaan Huygens studied how to compute logarithms by geometrical methods and calculated a quantity that, in retrospect, is the base-10 logarithm of, but he did not recognize itself as a quantity of interest.^[6]

The constant itself was introduced by Jacob Bernoulli in 1683, for solving the problem of continuous compounding of interest.^{[7] [8]} In his solution, the constant occurs as the limit$$\backslash lim\_\; \backslash left(1\; +\; \backslash frac\; \backslash right)^n,$$where represents the number of intervals in a year on which the compound interest is evaluated (for example,

n =12 for monthly compounding).

The first symbol used for this constant was the letter by Gottfried Leibniz in letters to Christiaan Huygens in 1690 and 1691.^[9]

Leonhard Euler started to use the letter for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons,^[10] and in a letter to Christian Goldbach on 25 November 1731.^{[11] [12]} The first appearance of in a printed publication was in Euler's Mechanica (1736).^[13] It is unknown why Euler chose the letter .^[14] Although some researchers used the letter in the subsequent years, the letter was more common and eventually became standard.^[15]

Euler proved that is the sum of the infinite series$$e\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots,$$where is the factorial of . The equivalence of the two characterizations using the limit and the infinite series can be proved via the binomial theorem.^[16]

Applications

Compound interest

Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest:

If the interest is credited twice in the year, the interest rate for each 6 months will be 50%, so the initial $1 is multiplied by 1.5 twice, yielding at the end of the year. Compounding quarterly yields, and compounding monthly yields . If there are compounding intervals, the interest for each interval will be and the value at the end of the year will be $1.00 × .

Bernoulli noticed that this sequence approaches a limit (the force of interest) with larger and, thus, smaller compounding intervals. Compounding weekly yields $2.692596..., while compounding daily yields $2.714567... (approximately two cents more). The limit as grows large is the number that came to be known as . That is, with continuous compounding, the account value will reach $2.718281828... More generally, an account that starts at $1 and offers an annual interest rate of will, after years, yield dollars with continuous compounding. Here, is the decimal equivalent of the rate of interest expressed as a percentage, so for 5% interest, .^[17]

Bernoulli trials

The number itself also has applications in probability theory, in a way that is not obviously related to exponential growth. Suppose that a gambler plays a slot machine that pays out with a probability of one in and plays it times. As increases, the probability that gambler will lose all bets approaches . For, this is already approximately 1/2.789509....

This is an example of a Bernoulli trial process. Each time the gambler plays the slots, there is a one in chance of winning. Playing times is modeled by the binomial distribution, which is closely related to the binomial theorem and Pascal's triangle. The probability of winning times out of trials is:^[18]

\Pr[k~ wins~of ~n] = \binom{n}{k} \left( (1)/(n) \right) ^{k\left(1} - (1)/(n) \right) ^{n-k} .

In particular, the probability of winning zero times is

\Pr[ 0 ~ wins~of ~n] = \left( 1 - (1)/(n) \right) ^{n} .

The limit of the above expression, as tends to infinity, is precisely .

Exponential growth and decay

Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself.^[19] Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). If the constant of proportionality is negative, then the quantity decreases over time, and is said to be undergoing exponential decay instead. The law of exponential growth can be written in different but mathematically equivalent forms, by using a different base, for which the number is a common and convenient choice:$$x(t)\; =\; x\_0\backslash cdot\; e^\; =\; x\_0\backslash cdot\; e^.$$Here,

x _{0} denotes the initial value of the quantity, is the growth constant, and \tau is the time it takes the quantity to grow by a factor of .

Standard normal distribution

See main article: Normal distribution.

The normal distribution with zero mean and unit standard deviation is known as the standard normal distribution, given by the probability density function$$\backslash phi(x)\; =\; \backslash frac\; e^.$$

The constraint of unit standard deviation (and thus also unit variance) results in the in the exponent, and the constraint of unit total area under the curve

\phi(x) results in the factor style 1 /\sqrt{ 2 \pi} . This function is symmetric around, where it attains its maximum value style 1 /\sqrt{ 2 \pi} , and has inflection points at .

Derangements

See main article: Derangement. Another application of, also discovered in part by Jacob Bernoulli along with Pierre Remond de Montmort, is in the problem of derangements, also known as the hat check problem:^[20] guests are invited to a party and, at the door, the guests all check their hats with the butler, who in turn places the hats into boxes, each labelled with the name of one guest. But the butler has not asked the identities of the guests, and so puts the hats into boxes selected at random. The problem of de Montmort is to find the probability that none of the hats gets put into the right box. This probability, denoted by

p _{n} , is:p _{n} = 1 - (1)/(1 !) + (1)/(2 !) - (1)/(3 !) + … + (( -1 ) ^{n})/(n!) = \sum ^{n} _{k =0} (( -1 ) ^{k})/(k!) .

As tends to infinity, approaches . Furthermore, the number of ways the hats can be placed into the boxes so that none of the hats are in the right box is rounded to the nearest integer, for every positive .^[21]

Optimal planning problems

The maximum value of

\sqrt[x]{x} occurs at x = e . Equivalently, for any value of the base, it is the case that the maximum value of x ^{-1} log _{b} x occurs at x = e (Steiner's problem, discussed below).

This is useful in the problem of a stick of length that is broken into equal parts. The value of that maximizes the product of the lengths is then either^[22]

n = \left\lfloor (L)/(e) \right\rfloor or \left\lceil (L)/(e) \right\rceil.

The quantity

x ^{-1} log _{b} x is also a measure of information gleaned from an event occurring with probability 1 /x (approximately 36 . 8 \% when x = e ), so that essentially the same optimal division appears in optimal planning problems like the secretary problem.

Asymptotics

The number occurs naturally in connection with many problems involving asymptotics. An example is Stirling's formula for the asymptotics of the factorial function, in which both the numbers and appear:^[23]

The principal motivation for introducing the number, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms.^[24] A general exponential has a derivative, given by a limit:

\begin{align} (d)/(dx) a ^{x} & = \lim _{h\to} (a ^{x+h} - a ^{x})/(h) = \lim _{h\to} (a ^{x} a ^{h} - a ^{x})/(h) \\ & = a ^{x} ⋅ \left(\lim _{h\to} (a ^{h} -1)/(h) \right). \end{align}

The parenthesized limit on the right is independent of the Its value turns out to be the logarithm of to base . Thus, when the value of is set this limit is equal and so one arrives at the following simple identity:

(d)/(dx) e ^{x} = e ^{x.}

Consequently, the exponential function with base is particularly suited to doing calculus. (as opposed to some other number) as the base of the exponential function makes calculations involving the derivatives much simpler.

Another motivation comes from considering the derivative of the base- logarithm (i.e.,), for :

\begin{align} (d)/(dx) log _{a} x & = \lim _{h\to} (log _{a(x} + h) -log _{a(x)})/(h) \\ & = \lim _{h\to} (log _{a(1} + h/x))/(x ⋅ h/x) \\ & = (1)/(x) log _{a\left(\lim} _{u\to} ( 1 + u) ^{(1)/(u)} \right) \\ & = (1)/(x) log _{a} e, \end{align}

where the substitution was made. The base- logarithm of is 1, if equals . So symbolically,

(d)/(dx) log _{e} x = (1)/(x) .

The logarithm with this special base is called the natural logarithm, and is denoted as ; it behaves well under differentiation since there is no undetermined limit to carry through the calculations.

Thus, there are two ways of selecting such special numbers . One way is to set the derivative of the exponential function equal to, and solve for . The other way is to set the derivative of the base logarithm to and solve for . In each case, one arrives at a convenient choice of base for doing calculus. It turns out that these two solutions for are actually the same: the number .

The Taylor series for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0:^[25] $$e^x\; =\; \backslash sum\_^\backslash infty\; \backslash frac.$$Setting

x = 1 recovers the definition of as the sum of an infinite series.

The natural logarithm function can be defined as the integral from 1 to

x of 1 /t , and the exponential function can then be defined as the inverse function of the natural logarithm. The number is the value of the exponential function evaluated at x = 1 , or equivalently, the number whose natural logarithm is 1. It follows that is the unique positive real number such that$$\backslash int\_1^e\; \backslash frac\; \backslash ,\; dt\; =\; 1.$$

Because is the unique function (up to multiplication by a constant) that is equal to its own derivative,

$$\backslash fracKe^x\; =\; Ke^x,$$

it is therefore its own antiderivative as well:^[26]

$$\backslash int\; Ke^x\backslash ,dx\; =\; Ke^x\; +\; C\; .$$

Equivalently, the family of functions

$$y(x)\; =\; Ke^x$$

where is any real or complex number, is the full solution to the differential equation

$$y\text{'}\; =\; y\; .$$

Inequalities

The number is the unique real number such thatfor all positive .^[27]

Also, we have the inequality$$e^x\; \backslash ge\; x\; +\; 1$$for all real, with equality if and only if . Furthermore, is the unique base of the exponential for which the inequality holds for all .^[28] This is a limiting case of Bernoulli's inequality.

Exponential-like functions

Steiner's problem asks to find the global maximum for the function

$$f(x)\; =\; x^\backslash frac\; .$$

This maximum occurs precisely at . (One can check that the derivative of is zero only for this value of .)

Similarly, is where the global minimum occurs for the function

$$f(x)\; =\; x^x\; .$$

The infinite tetration

x ^{x ^{x ^{⋅ ^{⋅ ^{⋅}}}}} or { ^{infty}x}

converges if and only if, shown by a theorem of Leonhard Euler.^{[29] [30] [31]}

Number theory

The real number is irrational. Euler proved this by showing that its simple continued fraction expansion does not terminate.^[32] (See also Fourier's proof that is irrational.)

Furthermore, by the Lindemann–Weierstrass theorem, is transcendental, meaning that it is not a solution of any non-zero polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873.^[33]

It is conjectured that is normal, meaning that when is expressed in any base the possible digits in that base are uniformly distributed (occur with equal probability in any sequence of given length).^[34]

In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. The constant is a period, but it is conjectured that is not.^[35]

Complex numbers

The exponential function may be written as a Taylor series

$$e^\; =\; 1\; +\; +\; +\; +\; \backslash cdots\; =\; \backslash sum\_^\; \backslash frac.$$

Because this series is convergent for every complex value of, it is commonly used to extend the definition of to the complex numbers.^[36] This, with the Taylor series for and , allows one to derive Euler's formula:

$$e^\; =\; \backslash cos\; x\; +\; i\backslash sin\; x,$$

which holds for every complex .^[36] The special case with is Euler's identity:

$$e^\; +\; 1\; =\; 0,$$which is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics. In addition, it is directly used in a proof that is transcendental, which implies the impossibility of squaring the circle.^{[37] [38]} Moreover, the identity implies that, in the principal branch of the logarithm,^[36]

$$\backslash ln\; (-1)\; =\; i\backslash pi\; .$$

Furthermore, using the laws for exponentiation,

$$(\backslash cos\; x\; +\; i\backslash sin\; x)^n\; =\; \backslash left(e^\backslash right)^n\; =\; e^\; =\; \backslash cos\; nx\; +\; i\; \backslash sin\; nx$$

for any integer, which is de Moivre's formula.

The expressions of and in terms of the exponential function can be deduced from the Taylor series:^[36] $$\backslash cos\; x\; =\; \backslash frac,\; \backslash qquad\; \backslash sin\; x\; =\; \backslash frac.$$

The expression$\backslash cos\; x\; +\; i\; \backslash sin\; x$is sometimes abbreviated as .^[39]

Representations

See main article: List of representations of {{mvar.

The number can be represented in a variety of ways: as an infinite series, an infinite product, a continued fraction, or a limit of a sequence. In addition to the limit and the series given above, there is also the continued fraction

e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , ..., 1 , 2 n, 1 , ...], ^[40]

which written out looks like

e = 2 + \cfrac{ 1 } { 1 + \cfrac{ 1 } { 2 + \cfrac{ 1 } { 1 + \cfrac{ 1 } { 1 + \cfrac{ 1 } { 4 + \cfrac{ 1 } { 1 + \cfrac{ 1 } { 1 + \ddots} } } } } } } .

The following infinite product evaluates to :^[22] $$e\; =\; \backslash frac\; \backslash left(\backslash frac\backslash right)^\; \backslash left(\backslash frac\backslash right)^\; \backslash left(\backslash frac\backslash right)^\; \backslash cdots.$$

Many other series, sequence, continued fraction, and infinite product representations of have been proved.

Stochastic representations

In addition to exact analytical expressions for representation of, there are stochastic techniques for estimating . One such approach begins with an infinite sequence of independent random variables, ..., drawn from the uniform distribution on [0, 1]. Let be the least number such that the sum of the first observations exceeds 1:

V = min \left\{ n \mid X _{1} + X _{2} + … + X _{n} > 1 \right\}.

Then the expected value of is : .^{[41] [42]}

Known digits

The number of known digits of has increased substantially during the last decades. This is due both to the increased performance of computers and to algorithmic improvements.^{[43] [44]}

Decimal digits	Computation performed by
1690	1	Jacob Bernoulli
1714	13	Roger Cotes^[45]
1748	23	Leonhard Euler^[46]
1853	137	William Shanks^[47]
1871	205	William Shanks^[48]
1884	346	J. Marcus Boorman^[49]
1949	2,010	John von Neumann (on the ENIAC)
1961	100,265	Daniel Shanks and John Wrench^[50]
1978	116,000	Steve Wozniak on the Apple II^[51]

Since around 2010, the proliferation of modern high-speed desktop computers has made it feasible for amateurs to compute trillions of digits of within acceptable amounts of time. On Dec 5, 2020, a record-setting calculation was made, giving to 31,415,926,535,897 (approximately) digits.^[52]

Computing the digits

One way to compute the digits of is with the series^[53] $$e=\backslash sum\_^\backslash infty\; \backslash frac.$$

A faster method involves two recursive functions

p(a,b) and q(a,b) . The functions are defined as

The expression $$1+\backslash frac$$ produces the th partial sum of the series above. This method uses binary splitting to compute with fewer single-digit arithmetic operations and thus reduced bit complexity. Combining this with fast Fourier transform-based methods of multiplying integers makes computing the digits very fast.

In computer culture

During the emergence of internet culture, individuals and organizations sometimes paid homage to the number .

In an early example, the computer scientist Donald Knuth let the version numbers of his program Metafont approach . The versions are 2, 2.7, 2.71, 2.718, and so forth.^[54]

In another instance, the IPO filing for Google in 2004, rather than a typical round-number amount of money, the company announced its intention to raise 2,718,281,828 USD, which is billion dollars rounded to the nearest dollar.^[55]

Google was also responsible for a billboard^[56] that appeared in the heart of Silicon Valley, and later in Cambridge, Massachusetts; Seattle, Washington; and Austin, Texas. It read ".com". The first 10-digit prime in is 7427466391, which starts at the 99th digit.^[57] Solving this problem and visiting the advertised (now defunct) website led to an even more difficult problem to solve, which consisted in finding the fifth term in the sequence 7182818284, 8182845904, 8747135266, 7427466391. It turned out that the sequence consisted of 10-digit numbers found in consecutive digits of whose digits summed to 49. The fifth term in the sequence is 5966290435, which starts at the 127th digit.^[58] Solving this second problem finally led to a Google Labs webpage where the visitor was invited to submit a résumé.^[59]

Further reading

• Maor, Eli; : The Story of a Number,
• Commentary on Endnote 10 of the book Prime Obsession for another stochastic representation
• McCartin, Brian J.. e: The Master of All. The Mathematical Intelligencer. 28. 2006. 2. 10–21. 10.1007/bf02987150. 123033482.

External links

• The number to 1 million places and NASA.gov 2 and 5 million places
• Approximations – Wolfram MathWorld
• Earliest Uses of Symbols for Constants Jan. 13, 2008
• "The story of ", by Robin Wilson at Gresham College, 28 February 2007 (available for audio and video download)
• Search Engine 2 billion searchable digits of, and

Notes and References

1. Web site: Weisstein. Eric W.. e. 2020-08-10. mathworld.wolfram.com. en. mathworld.
2. Book: The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics . illustrated . Clifford A. . Pickover . Sterling Publishing Company . 2009 . 978-1-4027-5796-9 . 166 . Extract of page 166
3. Book: Sawyer, W. W. . Mathematician's Delight . Penguin . 1961 . registration. 155 . en.
4. Book: Euler's Pioneering Equation: The most beautiful theorem in mathematics . illustrated . Robinn . Wilson . Oxford University Press . 2018 . 978-0-19-251405-9 . (preface) .
5. Book: Pi: A Biography of the World's Most Mysterious Number . illustrated . Alfred S. . Posamentier . Ingmar . Lehmann . Prometheus Books . 2004 . 978-1-59102-200-8 . 68 .
6. The Computation of Logarithms by Huygens . E. M. . Bruins . Constructive Function Theory . 1983 . 254–257.
7. Jacob Bernoulli considered the problem of continuous compounding of interest, which led to a series expression for . See: Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the Journal des Savants (Ephemerides Eruditorum Gallicanæ), in the year (anno) 1685.**), Acta eruditorum, pp. 219–23. On page 222, Bernoulli poses the question: "Alterius naturæ hoc Problema est: Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?" (This is a problem of another kind: The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would be owing [at the] end of [the] year?) Bernoulli constructs a power series to calculate the answer, and then writes: " … quæ nostra serie [mathematical expression for a geometric series] &c. major est. … si, debebitur plu quam & minus quam ." (… which our series [a geometric series] is larger [than]. … if, [the lender] will be owed more than and less than .) If, the geometric series reduces to the series for, so . (** The reference is to a problem which Jacob Bernoulli posed and which appears in the Journal des Sçavans of 1685 at the bottom of page 314.)
8. Book: Carl Boyer. Uta Merzbach. Uta Merzbach . A History of Mathematics. registration. 419. Wiley. 1991. 978-0-471-54397-8. 2nd.
9. Web site: Sämliche Schriften Und Briefe . Leibniz . Gottfried Wilhelm . 2003 . de . look for example letter nr. 6.
10. Euler, Meditatio in experimenta explosione tormentorum nuper instituta. Latin: Scribatur pro numero cujus logarithmus est unitas, e, qui est 2,7182817… (English: Written for the number of which the logarithm has the unit, e, that is 2,7182817...")
11. Lettre XV. Euler à Goldbach, dated November 25, 1731 in: P.H. Fuss, ed., Correspondance Mathématique et Physique de Quelques Célèbres Géomètres du XVIIIeme Siècle … (Mathematical and physical correspondence of some famous geometers of the 18th century), vol. 1, (St. Petersburg, Russia: 1843), pp. 56–60, see especially p. 58. From p. 58: " … (e denotat hic numerum, cujus logarithmus hyperbolicus est = 1), … " (… (e denotes that number whose hyperbolic [i.e., natural] logarithm is equal to 1) …)
12. Book: Remmert, Reinhold. Reinhold Remmert. Theory of Complex Functions. 136. Springer-Verlag. 1991. 978-0-387-97195-7.
13. Leonhard Euler, Mechanica, sive Motus scientia analytice exposita (St. Petersburg (Petropoli), Russia: Academy of Sciences, 1736), vol. 1, Chapter 2, Corollary 11, paragraph 171, p. 68. From page 68: Erit enim (dc)/(c) = (dy ds)/(rdx) seu c = e ^{\int (dy ds)/(rdx)} ubi denotat numerum, cuius logarithmus hyperbolicus est 1. (So it [i.e., {{mvar|c}}, the speed] will be (dc)/(c) = (dy ds)/(rdx) or c = e ^{\int (dy ds)/(rdx)} , where denotes the number whose hyperbolic [i.e., natural] logarithm is 1.)
14. Book: Calinger, Ronald . 2016 . Leonhard Euler: Mathematical Genius in the Enlightenment . Princeton University Press . 978-0-691-11927-4. p. 124.
15. Web site: Miller . Jeff . Earliest Uses of Symbols for Constants . MacTutor . University of St. Andrews, Scotland . 31 October 2023.
16. Book: Rudin, Walter . Walter Rudin . Principles of Mathematical Analysis . 3rd . McGraw–Hill . 1976 . 63–65 . 0-07-054235-X.
17. Book: Gonick , Larry . Larry Gonick . 2012 . The Cartoon Guide to Calculus . William Morrow . 978-0-06-168909-3 . 29–32.
18. Book: Kardar, Mehran . Mehran Kardar . Statistical Physics of Particles . Statistical Physics of Particles . 2007 . . 978-0-521-87342-0 . 860391091 . 41.
19. Book: https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions . 6.1 Exponential Functions . College Algebra 2e . OpenStax . Jay . Abramson . etal . 2023 . 978-1-951693-41-1.
20. Book: Grinstead. Charles M. . Snell. James Laurie . 1997 . Introduction to Probability . https://web.archive.org/web/20110727200156/http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.html . 2011-07-27 . published online under the GFDL. American Mathematical Society . 85 . 978-0-8218-9414-9 . J. Laurie Snell .
21. Book: Knuth, Donald . Donald Knuth . 1997 . The Art of Computer Programming . The Art of Computer Programming . I. Addison-Wesley. 183 . 0-201-03801-3.
22. Book: Mathematical constants. registration. Steven Finch. 2003. Cambridge University Press. 14. 978-0-521-81805-6.
23. Book: Gbur, Greg . Greg Gbur . 2011 . Mathematical Methods for Optical Physics and Engineering . 978-0-521516-10-5 . Cambridge University Press . 779.
24. Book: Kline, M. . 1998 . Calculus: An intuitive and physical approach . 337 ff . Dover Publications . 0-486-40453-6.
25. Book: Gilbert . Strang . Edwin . Herman . Calculus, volume 2 . OpenStax . etal . 6.3 Taylor and Maclaurin Series . https://openstax.org/books/calculus-volume-2/pages/6-3-taylor-and-maclaurin-series . 2023 . 978-1-947172-14-2.
26. Book: Gilbert . Strang . Edwin . Herman . Calculus, volume 2 . OpenStax . etal . 4.10 Antiderivatives . https://openstax.org/books/calculus-volume-1/pages/4-10-antiderivatives . 2023 . 978-1-947172-14-2.
27. Book: Dorrie. Heinrich. 100 Great Problems of Elementary Mathematics. 1965. Dover. 44–48.
28. A standard calculus exercise using the mean value theorem; see for example Apostol (1967) Calculus, § 6.17.41.
29. Euler, L. "De serie Lambertina Plurimisque eius insignibus proprietatibus." Acta Acad. Scient. Petropol. 2, 29–51, 1783. Reprinted in Euler, L. Opera Omnia, Series Prima, Vol. 6: Commentationes Algebraicae. Leipzig, Germany: Teubner, pp. 350–369, 1921. (facsimile)
30. Knoebel . R. Arthur . 1981 . Exponentials Reiterated . The American Mathematical Monthly . 88 . 4 . 235–252 . 10.2307/2320546 . 2320546 . 0002-9890.
31. Anderson . Joel . 2004 . Iterated Exponentials . The American Mathematical Monthly . 111 . 8 . 668–679 . 10.2307/4145040 . 4145040 . 0002-9890.
32. Web site: How Euler Did It: Who proved is Irrational?. Sandifer. Ed. Feb 2006. MAA Online. 2010-06-18. dead. https://web.archive.org/web/20140223072640/http://vanilla47.com/PDFs/Leonhard%20Euler/How%20Euler%20Did%20It%20by%20Ed%20Sandifer/Who%20proved%20e%20is%20irrational.pdf. 2014-02-23.
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