Editing Euler's formula (section)

==History==
In 1714, the English mathematician [[Roger Cotes]] presented a geometrical argument that can be interpreted (after correcting a misplaced factor of <math>\sqrt{-1}</math>) as:<ref>Cotes wrote:  ''"Nam si quadrantis circuli quilibet arcus, radio ''CE'' descriptus, sinun habeat ''CX'' sinumque complementi ad quadrantem ''XE'' ; sumendo radium ''CE'' pro Modulo, arcus erit rationis inter <math>EX + XC \sqrt{-1}</math>& ''CE'' mensura ducta in <math>\sqrt{-1}</math>."'' (Thus if any arc of a quadrant of a circle, described by the radius ''CE'', has sinus ''CX'' and sinus of the complement to the quadrant ''XE'' ; taking the radius ''CE'' as modulus, the arc will be the measure of the ratio between <math>EX + XC \sqrt{-1}</math> & ''CE'' multiplied by <math>\sqrt{-1}</math>.) That is, consider a circle having center ''E'' (at the origin of the (x,y) plane) and radius ''CE''.  Consider an angle ''θ'' with its vertex at ''E'' having the positive x-axis as one side and a radius ''CE'' as the other side.  The perpendicular from the point ''C'' on the circle to the x-axis is the "sinus" ''CX'' ; the line between the circle's center ''E'' and the point ''X'' at the foot of the perpendicular is ''XE'', which is the "sinus of the complement to the quadrant" or "cosinus".  The ratio between <math>EX + XC \sqrt{-1}</math> and ''CE'' is thus <math>\cos \theta + \sqrt{-1} \sin \theta \ </math>.  In Cotes' terminology, the "measure" of a quantity is its natural logarithm, and the "modulus" is a conversion factor that transforms a measure of angle into circular arc length (here, the modulus is the radius (''CE'') of the circle).  According to Cotes, the product of the modulus and the measure (logarithm) of the ratio, when multiplied by <math>\sqrt{-1}</math>, equals the length of the circular arc subtended by ''θ'', which for any angle measured in radians is ''CE'' • ''θ''.  Thus, <math>\sqrt{-1} CE \ln{\left ( \cos \theta + \sqrt{-1} \sin \theta \right ) \ } = (CE) \theta </math>.  This equation has a misplaced factor:  the factor of <math>\sqrt{-1}</math> should be on the right side of the equation, not the left side.  If the change of scaling by <math>\sqrt{-1}</math> is made, then, after dividing both sides by ''CE'' and exponentiating both sides, the result is:  <math>\cos \theta + \sqrt{-1} \sin \theta = e^{\sqrt{-1} \theta}</math>, which is Euler's formula.<br />
See:

* Roger Cotes (1714) "Logometria," ''Philosophical Transactions of the Royal Society of London'', '''29''' (338) : 5-45; see especially page 32.  Available on-line at:  [http://babel.hathitrust.org/cgi/pt?id=ucm.5324351035;view=2up;seq=38 Hathi Trust]
* Roger Cotes with Robert Smith, ed., ''Harmonia mensurarum'' … (Cambridge, England:  1722), chapter:  "Logometria",  [https://books.google.com/books?id=J6BGAAAAcAAJ&pg=PA28 p. 28].
* https://nrich.maths.org/1384</ref><ref name="Stillwell">{{cite book|author=John Stillwell|title=Mathematics and Its History|publisher=Springer|year=2002 |isbn=9781441960528| url = https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA315}}</ref><ref>Sandifer, C. Edward (2007), ''[https://books.google.com/books?id=sohHs7ExOsYC&pg=PA4 Euler's Greatest Hits]'', [[Mathematical Association of America]] {{ISBN|978-0-88385-563-8}}</ref>
<math display="block">ix = \ln(\cos x + i\sin x).</math>
Exponentiating this equation yields Euler's formula. Note that the logarithmic statement is not universally correct for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of {{math|2''πi''}}.
[[File:Rising circular.gif|thumb|Visualization of Euler's formula as a helix in three-dimensional space. The helix is formed by plotting points for various values of <math>\theta</math> and is determined by both the cosine and sine components of the formula. One curve represents the real component (<math>\cos\theta</math>) of the formula, while another curve, rotated 90 degrees around the z-axis (due to multiplication by <math>i</math>), represents the imaginary component (<math>\sin\theta</math>).]]
Around 1740 [[Leonhard Euler]] turned his attention to the exponential function and derived the equation named after him by comparing the series expansions of the exponential and trigonometric expressions.<ref>[[Leonhard Euler]] (1748) [http://www.17centurymaths.com/contents/euler/introductiontoanalysisvolone/ch8vol1.pdf Chapter 8: On transcending quantities arising from the circle] of [[Introduction to the Analysis of the Infinite]], page 214, section 138 (translation by Ian Bruce, pdf link from 17 century maths).</ref><ref name="Stillwell"/> The formula was first published in 1748 in his foundational work ''[[Introductio in analysin infinitorum]]''.<ref>Conway & Guy, pp. 254–255</ref>

[[Johann Bernoulli]] had found that<ref>{{cite journal|first=Johann |last=Bernoulli |title=Solution d'un problème concernant le calcul intégral, avec quelques abrégés par rapport à ce calcul |trans-title=Solution of a problem in integral calculus with some notes relating to this calculation |journal=Mémoires de l'Académie Royale des Sciences de Paris |pages=289–297|volume=1702 |date=1702}}</ref>
<math display="block">\frac{1}{1 + x^2} = \frac 1 2 \left( \frac{1}{1 - ix} + \frac{1}{1 + ix}\right).</math>

And since
<math display="block">\int \frac{dx}{1 + ax} = \frac{1}{a} \ln(1 + ax) + C,</math>
the above equation tells us something about [[complex logarithm]]s by relating natural logarithms to imaginary (complex) numbers. Bernoulli, however, did not evaluate the integral.

Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand [[complex logarithm]]s. Euler also suggested that complex logarithms can have infinitely many values.

The view of complex numbers as points in the [[complex plane]] was described about 50 years later by [[Caspar Wessel]].