Download e-book for iPad: An analysis of the Lanczos Gamma approximation by Pugh G.R.

By Pugh G.R.

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But there are many other functions which interpolate n! between the non-negative integers; why are the standard definitions the “right” ones? The life of gamma is retraced in the very entertaining paper of Davis [6], according to which the year 1729 saw “the birth” of the gamma function as Euler studied the pattern 1, 1 · 2, 1 · 2 · 3, . . The problem was simple enough: it was well known that interpolating formulas of the form 1+2+···+n = n(n + 1) 2 existed for sums; was there a similar formula f (n) = 1 · 2 · · · n for products?

1 Closed Form Formulas We begin by finding explicit expressions for v and fE,r in terms of Lambert W functions W0 and W−1 . This is useful for several reasons: the first is that graphs of v and fE,r can be easily plotted using the built in Lambert W evaluation routines of Maple 8. The second is that the 52 Chapter 4. The Functions v and fE,r smoothness properties of v and fE,r can be deduced directly from those of W0 and W−1 . Third, the expression for fE,r can be used to compute approximate values of the coefficients ak (r) using finite Fourier series.

9) and recalling the infinite 28 Chapter 2. A Primer on the Gamma Function product representation of the sine function ∞ 1− sin z = z k=1 z2 k2π2 , we arrive at the very useful reflection formula Γ(1 + z)Γ(1 − z) = πz . 10) is of great practical importance since it reduces the problem of computing the gamma function at arbitrary z to that of z with positive real part. 5 Stirling’s Series and Formula Among the well known preliminary results on the gamma function, one is of particular importance computationally and requires special attention: Stirling’s series.

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An analysis of the Lanczos Gamma approximation by Pugh G.R.


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