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Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand (1 + x)e x as a Taylor series in x, we use the known Taylor series of function e x:
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.
For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. [1] There are several versions of Taylor's theorem, some ...
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.
Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center c is equal to zero, for instance for Maclaurin series.
The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x.
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.
Furthermore, every polynomial is its own Maclaurin series. The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x 0 (as in the definition) but for all values of x (real or complex). The trigonometric functions, logarithm, and the power functions are analytic on any open set of their ...