The most general form of "an" exponential function is a power-law function of the form
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(1) |
where , , and are real numbers, is a positive real number, and is a real variable. When is positive, is an exponentially increasing function and when is negative, is an exponentially decreasing function.
In contrast, "the" exponential function (in elementary contexts sometimes called the "natural exponential function") is the function defined by
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(2) |
where e is positive real number is the base of the natural logarithm. The function is also the unique solution of the differential equation with initial condition . In other words, the exponential function is its own derivative, so
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(3) |
The exponential function defined for complex variable is an entire function in the complex plane.
The exponential function is implemented in the Wolfram Language as Exp[z].
The "natural" and general exponential functions are related to one another by a simple scalings of the variable and multiplicative prefactors via the identity
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(4) |
where is the natural logarithm.
The exponential function has the simple Maclaurin series
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(5) |
where is a factorial, and satisfies the limit
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(6) |
The exponential function satisfies the identity
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(7) |
It is also related to trigonometric functions via the identities
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(8) | |||
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(9) | |||
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(10) | |||
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(11) |
where is the Gudermannian (Beyer 1987, p. 164; Zwillinger 1995, p. 485).
If ,
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(12) |
Similarly, if
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(13) |
then
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(14) | |||
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(15) | |||
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(16) |
The exponential function has continued fraction
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(17) |
(Wall 1948, p. 348).
The above plot shows the function (Trott 2004, pp. 165-166).
Integrals involving the exponential function include
(Borwein et al. 2004, p. 55).
See also
Cis, Complex Exponentiation, e, Euler Formula, Exponent, Exponent Laws, Exponential Decay, Exponential Growth, Exponential Ramp, Exponentially Decreasing Function, Exponentially Increasing Function, Fourier Transform--Exponential Function, Gudermannian, Natural Exponential Function, Phasor, Power, Sigmoid Function Explore this topic in the MathWorld classroom
Related Wolfram sites
http://functions.wolfram.com/ElementaryFunctions/Exp/
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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.
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Cite this as:
Weisstein, Eric W. "Exponential Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExponentialFunction.html