Search results
Results from the WOW.Com Content Network
has radius of convergence 1, and diverges for z = 1 but converges for all other points on the boundary. The function f(z) of Example 1 is the derivative of g(z). Example 3: The power series = has radius of convergence 1 and converges everywhere on the boundary absolutely.
This explains why the Taylor series of f(x) diverges for |x| > 1, i.e., the radius of convergence is 1 because the complexified function has a pole at distance 1 from the evaluation point 0 and no further poles within the open disc of radius 1 around the evaluation point.
Such a function, and its analytic continuations, is called the hypergeometric function. The case when the radius of convergence is 0 yields many interesting series in mathematics, for example the incomplete gamma function has the asymptotic expansion
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, [1] but remained relatively unknown until Hadamard rediscovered it. [2]
This concept is often contrasted with uniform convergence.To say that = means that {| () |:} =, where is the common domain of and , and stands for the supremum.That is a stronger statement than the assertion of pointwise convergence: every uniformly convergent sequence is pointwise convergent, to the same limiting function, but some pointwise convergent sequences are not uniformly convergent.
This implies that for any function f of any Hölder class α > 0, the Fourier series converges everywhere to f(x). It is also known that for any periodic function of bounded variation, the Fourier series converges. In general, the most common criteria for pointwise convergence of a periodic function f are as follows:
So the radius of convergence of any probability generating function must be at least 1, by Abel's theorem for power series with non-negative coefficients. Probabilities and expectations [ edit ]
Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complex-analytic. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is non-zero. If a function is analytic, then it is infinitely differentiable, but in ...