Taylor expansions mathematical methods

Taylor series expansions, as described above, provide a very general method for representing a large class of mathematical functions. For the special case of periodic functions, a powerful alternative method is expansion in an infinite sum of sines and cosines, known as a trigonometric series or Fourier series. A periodic function is one that repeats in value when its argument is increased by multiples of a constant L, called the period or wavelength. For example. 

Changes of potential in relation to a potential determined for a given point Pq can be determined using the method of expansion of function E (x, y) into the Taylor series. After appropriate mathematical transformations, it can be shown that... 

The solution procedure was as follows Linearize Eq. (7.2b) by assuming v = const = Q/rcR2. Here Q is volume productivity of the plasticizing unit which is constant during the entire filling process and determined by its plastication parameters. Then find a solution to Eq. (7.2b) by the well-known methods of mathematical physics, and substitute it into Eq. (7.1a). Linearization of Eq. (7.2a) is performed by the expansion of L /KT(r, t) into the Taylor series. From Eq. (7.2a) and condition Q = const, we find the expression for v(r) ... 

Three methods are commonly used to estimate this quantity (1) slopes from a plot of n versus f, (2) equal-area graphic differentiation, or (3) Taylor series expansion. For details on these, see a mathematics handbook. The derivatives as found by equal-area graphic differentiation and other pertinent data are shown in the following table ... 

A common procedure for solving this overdetermined system is the method of variation of parameters (also referred to in the mathematical literature as Gauss-Newton non-linear least squares algorithm) (Vanicek and Krakiwsky 1982), and this procedure is described in the following. As approximate values of coordinates x° are known a priori, by Taylor s series expansion of the function / about point x°. 

Physical theories often require mathematical approximations. When functions are expressed as polynomial series, approximations can be systematically improved by keeping terms of increasingly higher order. One of the most important expansions is the Taylor series, an expression of a function in terms of its derivatives. These methods show that a Gaussian distribution function is a second-order approximation to a binomial distribution near its peak. We will hnd this useful for random walks, which are used to interpret diffusion, thermal conduction, and polymer conformations. In the next chapter we develop additional mathematical tools. 

We first consider Newton s method, an iterative technique that is based on the use of Taylor series expansions. As Taylor series are used extensively in numerical mathematics, we briefly review their use. 

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