Subject: integrate ( exp (x^3), x, 0, 1) still wrong
From: Evgeniy Maevskiy
Date: Tue, 15 May 2012 14:44:36 +0000 (UTC)
I'v wrote the next lisp-script for asymptotic of log(gamma_incomplete(a,x)) when
(real) x\to+\infty. If a\in\mathbb{Z} result is exact.
About the method see:
M. Fedoruk, Asymptotic methods in analysis. Analysis. I, Encyclopaedia Math.
Sci., 13, Springer, Berlin, 1989
; lngi(a,x) = \ln \int_x^{\infty} t^{a-1} e^t dt
(defun lngi (a x)
(declare (type flonum x a))
(let ((u 0.0) (s 0.0) (lx (log x)) (fa (floor a)) (sgn -1))
(do ((k 1 (+ k 1)))
((>= k a) 'done)
(setf u (- (+ u (log (- a k))) lx)
s (+ s (log (+ 1.0 (exp (- u s)))))))
(if (> a fa)
(do ((k (+ 1 fa) (+ k 1)))
((> k (+ a x)) 'done)
(setf u (- (+ u (log (- k a))) lx)
s (+ s (log (+ 1.0 (* sgn (exp (- u s))))))
sgn (* -1 sgn))))
(setf s (+ s (- (* (- a 1) lx) x)))))