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An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A series or, redundantly, an infinite series, is an infinite sum. It is often represented as [8] [15] ... Matrix exponential. For the matrix exponential: ...
In mathematics, a power series (in one variable) is an infinite series of the form = = + + + … where represents the coefficient of the nth term and c is a constant called the center of the series. Power series are useful in mathematical analysis , where they arise as Taylor series of infinitely differentiable functions .
Part of a series of articles on the: mathematical constant e; Properties; Natural logarithm; Exponential function; Applications; compound interest; Euler's identity; Euler's formula; half-lives. exponential growth and decay; Defining e; proof that e is irrational; representations of e; Lindemann–Weierstrass theorem; People; John Napier ...
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 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 unital Banach algebra B. In this setting, e 0 = 1 , and e x is invertible with inverse e − x for any x in B .
The Maclaurin series of the exponential function is absolutely convergent for ... There is an analogue of the comparison test for infinite series of functions called ...
The Taylor series for the exponential function can be deduced from the facts that the exponential function is its own derivative and that it equals 1 when evaluated at 0: [29] = =!. Setting x = 1 {\displaystyle x=1} recovers the definition of e as the sum of an infinite series.