100 research outputs found
On the Power Series Expansion of the Reciprocal Gamma Function
Using the reflection formula of the Gamma function, we derive a new formula
for the Taylor coefficients of the reciprocal Gamma function. The new formula
provides effective asymptotic values for the coefficients even for very small
values of the indices. Both the sign oscillations and the leading order of
growth are given.Comment: Corrected a sign in equation (3.21) due to a minor error in (3.19)
where the fraction was inadvertently inverted. Now the rough approximation
provides an elementary proof that the order of the reciprocal gamma function
is 1 and that its type is maxima
Robust Padé approximation via SVD
Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to infinity, unlike traditional Padé approximants, which converge only in measure or capacity
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