Search results
Results from the WOW.Com Content Network
Exponential decay. A quantity undergoing exponential decay. Larger decay constants make the quantity vanish much more rapidly. This plot shows decay for decay constant ( λ) of 25, 5, 1, 1/5, and 1/25 for x from 0 to 5. A quantity is subject to exponential decay if it decreases at a rate proportional to its current value.
This behavior is referred to as a "decaying" exponential function. The time τ (tau) is referred to as the "time constant" and can be used (as in this case) to indicate how rapidly an exponential function decays. Here: t is time (generally t > 0 in control engineering) V 0 is the initial value (see "specific cases" below). Specific cases
Exponential growth is a process that increases quantity over time at an ever-increasing rate. It occurs when the instantaneous rate of change (that is, the derivative) of a quantity with respect to time is proportional to the quantity itself. Described as a function, a quantity undergoing exponential growth is an exponential function of time ...
It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group . Let X be an n×n real or complex matrix. The exponential of X, denoted by eX or exp (X), is the n×n matrix given by the power series.
where. (Note that the value of the expression is independent of the value of n, which is why it does not appear in the integral.) where. and Γ (x,y) is the upper incomplete gamma function. when , , and. when , , and.
greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term). 1, the progression is a constant sequence. between −1 and 1 but not zero, there will be exponential decay towards zero (→ 0). −1, the absolute value of each term in the sequence is constant and terms ...
The equivalent concept to doubling time for a material undergoing a constant negative relative growth rate or exponential decay is the half-life. The equivalent concept in base- e is e -folding . Graphs comparing doubling times and half lives of exponential growths (bold lines) and decay (faint lines), and their 70/ t and 72/ t approximations.
Matrix differential equation. A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to ...