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- Log ----------------------------------------------------------------- commit d17228b86105b8a6ac652459b1100266ad0311b3 Author: Raymond Toy toy.raymond@gmail.com Date: Thu Mar 17 22:14:25 2011 -0400

Add Fresnel integrals; fix issue in incomplete-gamma for large args.

o INCOMPLETE-GAMMA was returning bad values for large (complex) arguments. Fix this by using incomplete gamma tail function since the incomplete gamma function approaches gamma for large arguments.

o Implement Fresnel S and C functions.

diff --git a/qd-gamma.lisp b/qd-gamma.lisp index a25bc7d..dfc87a6 100644 --- a/qd-gamma.lisp +++ b/qd-gamma.lisp @@ -292,7 +292,9 @@ (if (< z (- a 1)) (cf-incomplete-gamma a z) (- (gamma a) (cf-incomplete-gamma-tail a z))) - (cf-incomplete-gamma a z)))) + (if (< (abs z) (abs a)) + (cf-incomplete-gamma a z) + (- (gamma a) (cf-incomplete-gamma-tail a z))))))

(defun erf (z) "Error function: @@ -341,3 +343,48 @@ ;; Wolfram gives E(v,z) = z^(v-1)*gamma_incomplete_tail(1-v,z) (* (expt z (- v 1)) (incomplete-gamma-tail (- 1 v) z))) + +(defun fresnel-s (z) + "Fresnel S: + + S(z) = integrate(sin(%pi*t^2/2), t, 0, z) " + (let ((sqrt-pi (sqrt (float-pi z)))) + (flet ((fs (z) + ;; Wolfram gives + ;; + ;; S(z) = (1+%i)/4*(erf(c*z) - %i*erf(conjugate(c)*z)) + ;; + ;; where c = sqrt(%pi)/2*(1+%i). + (* #c(1/4 1/4) + (- (erf (* #c(1/2 1/2) sqrt-pi z)) + (* #c(0 1) + (erf (* #c(1/2 -1/2) sqrt-pi z))))))) + (if (realp z) + ;; FresnelS is real for a real argument. And it is odd. + (if (minusp z) + (- (realpart (fs (- z)))) + (realpart (fs z))) + (fs z))))) + +(defun fresnel-c (z) + "Fresnel C: + + C(z) = integrate(cos(%pi*t^2/2), t, 0, z) " + (let ((sqrt-pi (sqrt (float-pi z)))) + (flet ((fs (z) + ;; Wolfram gives + ;; + ;; C(z) = (1-%i)/4*(erf(c*z) + %i*erf(conjugate(c)*z)) + ;; + ;; where c = sqrt(%pi)/2*(1+%i). + (* #c(1/4 -1/4) + (+ (erf (* #c(1/2 1/2) sqrt-pi z)) + (* #c(0 1) + (erf (* #c(1/2 -1/2) sqrt-pi z))))))) + (if (realp z) + ;; FresnelS is real for a real argument. And it is odd. + (if (minusp z) + (- (realpart (fs (- z)))) + (realpart (fs z))) + (fs z))))) + \ No newline at end of file

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Summary of changes: qd-gamma.lisp | 49 ++++++++++++++++++++++++++++++++++++++++++++++++- 1 files changed, 48 insertions(+), 1 deletions(-)

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