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A power series with a positive radius of convergence can be made into a holomorphic function by taking its argument to be a complex variable. The radius of convergence can be characterized by the following theorem: The radius of convergence of a power series f centered on a point a is equal to the distance from a to the nearest point where f ...
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 .
An analogous statement for convergence of improper integrals is proven using integration by parts. If the integral of a function f is uniformly bounded over all intervals , and g is a non-negative monotonically decreasing function , then the integral of fg is a convergent improper integral.
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
The argument, first given by Cauchy, hinges on Cauchy's integral formula and the power series expansion of the expression 1 w − z . {\displaystyle {\frac {1}{w-z}}.} Let D {\displaystyle D} be an open disk centered at a {\displaystyle a} and suppose f {\displaystyle f} is differentiable everywhere within an open neighborhood containing the ...
Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately).
The utility of Abel's theorem is that it allows us to find the limit of a power series as its argument (that is, ) approaches from below, even in cases where the radius of convergence, , of the power series is equal to and we cannot be sure whether the limit should be finite or not.
This expression is found by writing 2 s Li s (−z) / (−z) = Φ(z 2, s, 1 ⁄ 2) − z Φ(z 2, s, 1), where Φ is the Lerch transcendent, and applying the Abel–Plana formula to the first Φ series and a complementary formula that involves 1 / (e 2πt + 1) in place of 1 / (e 2πt − 1) to the second Φ series. We can express an integral for ...