Exponential Function

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The most general form of "an" exponential function is a power-law function of the form

where 👁 a
, 👁 c
, and 👁 d
are real numbers, 👁 b
is a positive real number, and 👁 x
is a real variable. When 👁 c
is positive, 👁 f(x)
is an exponentially increasing function and when 👁 c
is negative, 👁 f(x)
is an exponentially decreasing function.

In contrast, "the" exponential function (in elementary contexts sometimes called the "natural exponential function") is the function defined by

where e is positive real number 👁 e=2.718...
is the base of the natural logarithm. The function 👁 exp(x)
is also the unique solution of the differential equation 👁 df/dx=f(x)
with initial condition 👁 f(0)=1
. In other words, the exponential function is its own derivative, so

The exponential function 👁 exp(z)=e^z
defined for complex variable 👁 z
is an entire function in the complex plane.

The exponential function is implemented in the Wolfram Language as [z].

The "natural" and general exponential functions are related to one another by a simple scalings of the variable 👁 x
and multiplicative prefactors via the identity

where 👁 lnz
is the natural logarithm.

The exponential function has the simple Maclaurin series

where 👁 n!
is a factorial, and satisfies the limit

The exponential function satisfies the identity

It is also related to trigonometric functions via the identities

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

If 👁 z=x+iy
,

Similarly, if

then

The exponential function has continued fraction

(Wall 1948, p. 348).

The above plot shows the function 👁 e^(1/z)
(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 👁 exp(bx+c)
" and "Exponentials of Powers 👁 exp(-ax^nu)
." 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.

Referenced on Wolfram|Alpha

Exponential Function

Cite this as:

Weisstein, Eric W. "Exponential Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExponentialFunction.html

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