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The number r is maximal in the following sense: there always exists a complex number x with | x − c | = r such that no analytic continuation of the series can be defined at x. The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.
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 formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
Alternatively, the equality can be justified by multiplying the power series on the left by 1 − x, and checking that the result is the constant power series 1 (in other words, that all coefficients except the one of x 0 are equal to 0). Moreover, there can be no other power series with this property.
Z tables use at least three different conventions: Cumulative from mean gives a probability that a statistic is between 0 (mean) and Z. Example: Prob(0 ≤ Z ≤ 0.69) = 0.2549. Cumulative gives a probability that a statistic is less than Z. This equates to the area of the distribution below Z. Example: Prob(Z ≤ 0.69) = 0.7549. Complementary ...
In order for the solution method to work, as in linear equations, it is necessary to express every term in the nonlinear equation as a power series so that all of the terms may be combined into one power series. As an example, consider the initial value problem ″ + ′ + ′ =; = , ′ = which describes a solution to capillary-driven flow in ...
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If a(n) is a discrete function of a positive integer n, then the power series associated to a(n) is the series = where x is a real variable (see Z-transform). Replacing summation over n with integration over t , a continuous version of the power series becomes ∫ 0 ∞ f ( t ) x t d t {\displaystyle \int _{0}^{\infty }f(t)x^{t}\,dt} where the ...