Exponential Function

Min Max

Min   Max       Re       Im

The exponential function is the entire function defined by

(1)

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

(2)

If ,

(3)

The exponential function satisfies the identities

			(4)
			(5)
			(6)
			(7)

where  is the Gudermannian (Beyer 1987, p. 164; Zwillinger 1995, p. 485).

The exponential function has Maclaurin series

(8)

and satisfies the limit

(9)

If

(10)

then

			(11)
			(12)
			(13)

The exponential function has continued fraction

(14)

(Wall 1948, p. 348).

Min   Max       Re       Im

The above plot shows the function  (Trott 2004, pp. 165-166).

Integrals involving the exponential function include

			(15)
			(16)

(Borwein et al. 2004, p. 55).

REFERENCES:

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.