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A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ ( lambda) is a positive rate called the exponential decay constant, disintegration constant, [1] rate constant, [2] or transformation constant: [3]
Half-life is constant over the lifetime of an exponentially decaying quantity, and it is a characteristic unit for the exponential decay equation. The accompanying table shows the reduction of a quantity as a function of the number of half-lives elapsed.
The stretched exponential function. is obtained by inserting a fractional power law into the exponential function. In most applications, it is meaningful only for arguments t between 0 and +∞. With β = 1, the usual exponential function is recovered. With a stretching exponent β between 0 and 1, the graph of log f versus t is ...
Weissberger’s modified exponential decay model, or simply, Weissberger’s model, is a radio wave propagation model that estimates the path loss due to the presence of one or more trees in a point-to-point telecommunication link.
In this distribution, the exponential decay term eventually overwhelms the power-law behavior at very large values of . This distribution does not scale [further explanation needed] and is thus not asymptotically as a power law; however, it does approximately scale over a finite region before the cutoff.
The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals .
The doubling time is a characteristic unit (a natural unit of scale) for the exponential growth equation, and its converse for exponential decay is the half-life .
Geometric sequences (with common ratio not equal to −1, 1 or 0) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as 4, 15, 26, 37, 48, … (with common difference 11). This result was taken by T.R. Malthus as the mathematical foundation of his Principle of Population .