Search results
Results from the WOW.Com Content Network
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 .
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).
Two cases arise: The first case is theoretical: when you know all the coefficients then you take certain limits and find the precise radius of convergence.; The second case is practical: when you construct a power series solution of a difficult problem you typically will only know a finite number of terms in a power series, anywhere from a couple of terms to a hundred terms.
The power series method will give solutions only to initial value problems (opposed to boundary value problems), this is not an issue when dealing with linear equations since the solution may turn up multiple linearly independent solutions which may be combined (by superposition) to solve boundary value problems as well. A further restriction ...
For any , the coefficient of /! in the moment generating function (expressed as an exponential power series in ) is the normal distribution's expected value [] . The cumulant generating function is the logarithm of the moment generating function, namely
and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centered at b if and only if its Taylor series converges to the value of the function at each point of the disk. If f (x) is equal to the sum of its Taylor series for all x in the complex plane, it is called entire.
The solution () / has a power series starting with the power zero. In a power series starting with the recurrence relation places no restriction on the coefficient for the term , which can be set arbitrarily. If it is set to zero then with this differential equation all the other coefficients will be zero and we obtain the solution 1/z.
The lower exponent-p central series is sometimes simply called the p-central series. There is a unique most quickly ascending such series, the upper exponent-p central series S defined by: S 0 (G) = 1 S n+1 (G)/S n (G) = Ω(Z(G/S n (G))) where Ω(Z(H)) denotes the subgroup generated by (and equal to) the set of central elements of H of order ...