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oct
oct
Commits
c7fc989d
Commit
c7fc989d
authored
Apr 08, 2012
by
Raymond Toy
Browse files
Fix typo in %big-a. Just use incomplete-gamma-tail in big-i.
parent
95aa580c
Changes
1
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Inline
Side-by-side
qd-bessel.lisp
View file @
c7fc989d
...
...
@@ -298,7 +298,7 @@
(
cond
((
and
(
=
m
v
)
(
minusp
m
))
(
if
(
<
n
m
)
(
%big-a
n
v
)
(
let
((
result
(
%big-a
(
+
n
m
)
v
)))
(
let
((
result
(
%big-a
(
+
n
m
)
(
-
v
)
)))
(
if
(
oddp
(
truncate
m
))
result
(
-
result
)))))
...
...
@@ -310,32 +310,18 @@
;;
;; Use the substitution u=1+s to get a new integral
;;
;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf)
;; = exp(t*z) * integrate(u^(-v-2*n)*exp(-t*u*z), u, 1, inf)
;; = exp(t*z)*t^(v+2*n-1)*z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z)
;;
;; The continued fraction for incomplete_gamma_tail(a,z) is
;;
;; z^a*exp(-z)/CF
;;
;; So incomplete_gamma_tail(1-v-2*n, t*z) is
;;
;; (t*z)^(1-v-2*n)/CF
;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf)
;; = exp(t*z) * integrate(u^(-v-2*n)*exp(-t*u*z), u, 1, inf)
;; = exp(t*z)*t^(v+2*n-1)*z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z)
;;
;;
which finally gives
;;
Thus,
;;
;; integrate(exp(-t*z*s)*(1+s)^(-2*n-v), s, 0, inf)
;; = (t*z)^(1-v-2*n)/CF
;; I[n](t, z, v) = z^(v+2*n-1)*incomplete_gamma_tail(1-v-2*n,t*z)
;;
;; and I[n](t, z, v) = exp(-t*z)/CF
(
defun
big-i
(
n
theta
z
v
)
(
/
(
exp
(
-
(
*
theta
z
)))
(
let*
((
a
(
-
1
v
n
n
))
(
z-a
(
-
(
*
theta
z
)
a
)))
(
lentz
#'
(
lambda
(
n
)
(
+
n
n
1
z-a
))
#'
(
lambda
(
n
)
(
*
n
(
-
a
n
)))))))
(
let*
((
a
(
-
1
v
n
n
)))
(
*
(
expt
z
(
-
a
))
(
incomplete-gamma-tail
a
(
*
theta
z
)))))
(
defun
sum-big-ia
(
big-n
v
z
)
)
...
...
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