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Explicitly for any real constant Exponential function on the complex plane.The transition from dark to light colors shows that the magnitude of the exponential function is increasing to the right.The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any Banach algebra is the Lie algebra of the Lie group of all positive real numbers under multiplication, the ordinary exponential function for real arguments is a special case of the Lie algebra situation.In 1941, New York intellectual playwright Barton Fink comes to Hollywood to write a Wallace Beery wrestling picture.The exponential function arises whenever a quantity grows or decays at a rate proportional to its current value.One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 This is one of a number of characterizations of the exponential function; others involve series or differential equations.

If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth (see Malthusian catastrophe), continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time. As functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function (i.e., its derivative) is directly proportional to the value of the function.The constant of proportionality of this relationship is the natural logarithm of the base Since changing the base of the exponential function merely results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function",.From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, .The derivative (rate of change) of the exponential function is the exponential function itself.

The most common definition of the complex exponential function parallels the power series definition for real arguments, where the real variable is replaced by a complex one: Termwise multiplication of two copies of these power series in the Cauchy sense, permitted by Mertens' theorem, shows that the defining multiplicative property of exponential functions continues to hold for all complex arguments: In this expansion, the rearrangement of the terms into real and imaginary parts is justified by the absolute convergence of the series.

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