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Since a second-order expansion for [()] has already been derived above, it only remains to find [() ()]. Treating () as a two-variable function, the second-order Taylor expansion is as follows:
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:
In fact, for a smooth enough function, we have the similar Taylor expansion (+) = | | ()! + (,), where the last term (the remainder) depends on the exact version of Taylor's formula.
Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, [2] although an earlier version of the result was already mentioned in 1671 by James Gregory. [3] Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis.
Thus to -approximate () = using a polynomial with lowest degree 3, we do so for () with < / by truncating its Taylor expansion. Now iterate this construction by plugging in the lowest-degree-3 approximation into the Taylor expansion of g ( x ) {\displaystyle g(x)} , obtaining an approximation of lowest degree 9, 27, 81...
The theorem was proved by Lagrange [2] and generalized by Hans Heinrich Bürmann, [3] [4] [5] both in the late 18th century. There is a straightforward derivation using complex analysis and contour integration ; [ 6 ] the complex formal power series version is a consequence of knowing the formula for polynomials , so the theory of analytic ...
The shift operator acting on functions of a real variable is a unitary operator on (). In both cases, the (left) shift operator satisfies the following commutation relation with the Fourier transform: F T t = M t F , {\displaystyle {\mathcal {F}}T^{t}=M^{t}{\mathcal {F}},} where M t is the multiplication operator by exp( itx ) .
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