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A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
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.
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.
Asymptotic series, typically called asymptotic expansions, are infinite series whose terms are functions of a sequence of different asymptotic orders and whose partial sums are approximations of some other function in an asymptotic limit. In general they do not converge, but they are still useful as sequences of approximations, each of which ...
In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
A Fourier series (/ ˈ f ʊr i eɪ,-i ər / [1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are ...
This is the function series expansion, integral and integration formula for a double integral whose integrand's series expansion contains consecutive integer exponents. [ 10 ] f ( y 1 , y 2 ) = ∑ n = 0 ∞ ( − 1 ) n 1 n 1 !
Each term of this modified series is a rational function with its poles at = in the complex plane, the same place where the arctangent function has its poles. By contrast, a polynomial such as the Taylor series for arctangent forces all of its poles to infinity.