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Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab
English: Graph of the complex exponential function in v, and w dimensions where v+iw = exp(x+iy) for values of x from -1.5 to 1.5 and values of y from -π to π. Areas checkered with green are for positive values of x.
Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.
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 characteristically stretched, hence the name of the function. The compressed exponential function (with β > 1) has less practical importance, with the notable exception of β = 2, which gives the normal ...
In mathematics, the exponential function can be characterized in many ways. This article presents some common characterizations, discusses why each makes sense, and proves that they are all equivalent. The exponential function occurs naturally in many branches of mathematics. Walter Rudin called it "the most important function in mathematics". [1]
The definition of e x as the exponential function allows defining b x for every positive real numbers b, in terms of exponential and logarithm function. Specifically, the fact that the natural logarithm ln(x) is the inverse of the exponential function e x means that one has = () = for every b > 0.
A double exponential function (red curve) compared to a single exponential function (blue curve). A double exponential function is a constant raised to the power of an exponential function . The general formula is f ( x ) = a b x = a ( b x ) {\displaystyle f(x)=a^{b^{x}}=a^{(b^{x})}} (where a >1 and b >1), which grows much more quickly than an ...
p-adic function: a function whose domain is p-adic. Linear function; also affine function. Convex function: line segment between any two points on the graph lies above the graph. Also concave function. Arithmetic function: A function from the positive integers into the complex numbers. Analytic function: Can be defined locally by a convergent ...