Editing Transcendental function (section)

==History==
The transcendental functions [[sine]] and [[cosine]] were [[trigonometric tables|tabulated]] from physical measurements in antiquity, as evidenced in Greece ([[Hipparchus]]) and India ([[jya]] and [[koti-jya]]). In describing [[Ptolemy's table of chords]], an equivalent to a table of sines, [[Olaf Pedersen]] wrote:

{{quote|The mathematical notion of continuity as an explicit concept is unknown to Ptolemy. That he, in fact, treats these functions as continuous appears from his unspoken presumption that it is possible to determine a value of the dependent variable corresponding to any value of the independent variable by the simple process of [[linear interpolation]].<ref>{{cite book |author-link=Olaf Pedersen |first=Olaf |last=Pedersen |title=Survey of the Almagest |publisher=[[Odense University Press]] |location= |date=1974 |isbn=87-7492-087-1 |pages=84 |url=}}</ref>}}

A revolutionary understanding of these [[circular function]]s occurred in the 17th century and was explicated by [[Leonhard Euler]] in 1748 in his [[Introduction to the Analysis of the Infinite]]. These ancient transcendental functions became known as [[continuous function]]s through [[quadrature (mathematics)|quadrature]] of the [[rectangular hyperbola]] {{math|1=''xy'' = 1}} by [[Grégoire de Saint-Vincent]] in 1647, two millennia after [[Archimedes]] had produced ''[[The Quadrature of the Parabola]]''. 

The area under the [[hyperbola]] was shown to have the scaling property of constant area for a constant ratio of bounds. The [[hyperbolic logarithm]] function so described was of limited service until 1748 when [[Leonhard Euler]] related it to functions where a constant is raised to a variable exponent, such as the [[exponential function]] where the constant [[base (exponentiation)|base]] is [[e (mathematical constant)|e]]. By introducing these transcendental functions and noting the [[bijection]] property that implies an [[inverse function]], some facility was provided for algebraic manipulations of the [[natural logarithm]] even if it is not an algebraic function.

The exponential function is written {{nowrap|<math> \exp (x) = e^x</math>.}} Euler identified it with the [[infinite series]] {{nowrap|<math display="inline">\sum_{k=0} ^{\infty} x^k / k ! </math>,}} where {{math|''k''!}} denotes the [[factorial]] of {{mvar|k}}.

The even and odd terms of this series provide sums denoting {{math|cosh(''x'')}} and {{math|sinh(''x'')}}, so that <math>e^x = \cosh x + \sinh x.</math> These transcendental [[hyperbolic function]]s can be converted into circular functions sine and cosine by introducing {{math|(−1)<sup>''k''</sup>}} into the series, resulting in [[alternating series]]. After Euler, mathematicians view the sine and cosine this way to relate the transcendence to logarithm and exponent functions, often through [[Euler's formula]] in [[complex number]] arithmetic.