Exponential Function
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The exponential function is the entire function defined by
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(1)
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where e is the solution of the equation 
so that 
. 
is also the unique solution of the equation
with 
.
The exponential function is implemented in the Wolfram Language as Exp[z].
It satisfies the identity
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(2)
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If 
,
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(3)
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The exponential function satisfies the identities
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(4)
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(5)
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(6)
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(7)
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where 
is the Gudermannian
(Beyer 1987, p. 164; Zwillinger 1995, p. 485).
The exponential function has Maclaurin series
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(8)
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and satisfies the limit
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(9)
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If
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(10)
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then
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(11)
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 |  | ![ln{bcsc[tan^(-1)(b/a)]}](/images/equations/ExponentialFunction/Inline25.gif) |
(12)
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 |  | ![ln{asec[tan^(-1)(b/a)]}.](/images/equations/ExponentialFunction/Inline28.gif) |
(13)
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The exponential function has continued fraction
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(14)
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(Wall 1948, p. 348).
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The above plot shows the function 
(Trott 2004,
pp. 165-166).
Integrals involving the exponential function include
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(15)
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(16)
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(Borwein et al. 2004, p. 55).
RELATED WOLFRAM SITES: http://functions.wolfram.com/ElementaryFunctions/Exp/
Abramowitz, M. and Stegun, I. A. (Eds.). "Exponential Function." §4.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 69-71, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 217, 1987.
Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.
Finch, S. "Linear Independence of Exponential Functions." http://algo.inria.fr/csolve/sstein.html.
Fischer, G. (Ed.). Plates 127-128 in Mathematische Modelle aus den Sammlungen von Universitäten und Museen, Bildband. Braunschweig, Germany: Vieweg, pp. 124-125, 1986.
Krantz, S. G. "The Exponential and Applications." §1.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 7-12, 1999.
Spanier, J. and Oldham, K. B. "The Exponential Function 
"
and "Exponentials of Powers 
."
Chs. 26-27 in An
Atlas of Functions. Washington, DC: Hemisphere, pp. 233-261, 1987.
Trott, M. "Elementary Transcendental Functions." §2.2.3 in The Mathematica GuideBook for Programming. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.
Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.
Yates, R. C. "Exponential Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 86-97, 1952.
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.
Weisstein, Eric W. "Exponential Function." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ExponentialFunction.html
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