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From: SourceForge.net <noreply@so...>  20090926 23:30:08

Bugs item #2867727, was opened at 20090927 01:30 Message generated for change (Tracker Item Submitted) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867727&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None Status: Open Resolution: None Priority: 5 Private: No Submitted By: Dieter Kaiser (crategus) Assigned to: Nobody/Anonymous (nobody) Summary: specint: wrong result for Parabolic Cylinder D function Initial Comment: A look at the implemented algorithm for the Laplace transform of a hypergeometric function shows that the integral for the Parabolic Cylinder D function should work only for an argument t^k with an exponent k < 1, e.g with an argument sqrt(t) (I have already introduced a new symbol in my sandbox for the Parabolic Cylinder D function): (%i2) factor(ratsimp(specint(exp(s*t)*parabolic_cylinder_d(0,sqrt(t)),t))); (%o2) 4/(4*s+1) This is a correct result. We can verify the result using a specific expansion: parabolic_cylinder_d(0,z) = exp(z^2/4) parabolic_cylinder_d(1,z) = z*exp(z^2/4) parabolic_cylinder_d(2,z) = (z^21)*exp(z^2/4) Now some more results with the first expansion for v=0: (%i3) d0(z):=exp(z^2/4)$ We can verify the result for the Parabolic Cylinder D function with an argument sqrt(t): (%i6) factor(ratsimp(specint(exp(s*t)*d0(sqrt(t)),t))); (%o6) 4/(4*s+1) Now we do the integration with an argument t using the expansion. We get a result which contains the Error function: (%i7) factor(ratsimp(specint(exp(s*t)*d0(t),t))); (%o7) sqrt(%pi)*%e^s^2*(erf(s)1) The following integral should not work, but it does and the answer is wrong: (%i8) factor(ratsimp(specint(exp(s*t)*parabolic_cylinder_d(0,t),t))); (%o8) 2^(3/2)*gamma(3/4)/(4*s+1)^(3/4) The routine f16p217test does not fail as expected and produces the wrong result. Remarks: 1. $specint uses the Parabolic Cylinder D function and its Laplace transform for the Hermite and the Erfc function. For the Hermite function the algorithm does not work as expected. For the erfc function it is more simple to integrate (1erf(z)). 2. The Parabolic Cylinder D function is transformed in terms of the Whittaker W function. In a second step the Whittaker W function is transformed to a hypergeometric representation. This representation is integrated. But we already have a routine simpdtf which does the transformation to a hypergeometric representation in one step. Dieter Kaiser  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867727&group_id=4933 
From: SourceForge.net <noreply@so...>  20090926 18:53:03

Bugs item #2867499, was opened at 20090926 18:02 Message generated for change (Settings changed) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867499&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None >Status: Closed >Resolution: Fixed Priority: 5 Private: No Submitted By: Dieter Kaiser (crategus) Assigned to: Nobody/Anonymous (nobody) Summary: specint(exp(s*t)*CONST*t^2*bessel_y(1,t),t) wrong Initial Comment: Maxima does not take into a account a constant part of an integrand when doing the Laplace transform of bessel_y(v,a*t) where v is an integer order. The symbol CONST is missing in the following result: (%i5) ratsimp(specint(exp(s*t)*CONST*t^2*bessel_y(1,t),t)); (%o5) (3*s*log((sqrt(s^2+1)+s)/(sqrt(s^2+1)s))+sqrt(s^2+1)*(2*s^24)) /(%pi*sqrt(s^2+1)*(s^4+2*s^2+1)) The error is in the routine f2p105v2cond. The constant part is extracted from the variable l but not multiplied to the result of the Laplace transform. Dieter Kaiser  >Comment By: Dieter Kaiser (crategus) Date: 20090926 20:53 Message: Fixed in hypgeo.lisp revision 1.66. Closing this bug report. Dieter Kaiser  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867499&group_id=4933 
From: SourceForge.net <noreply@so...>  20090926 18:52:20

Bugs item #2867434, was opened at 20090926 15:35 Message generated for change (Settings changed) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867434&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None >Status: Closed >Resolution: Fixed Priority: 5 Private: No Submitted By: Dieter Kaiser (crategus) Assigned to: Nobody/Anonymous (nobody) Summary: specint(exp(s*t)*t^2*%h[1,1](t),t) does not work Initial Comment: We know hankel_1(v,z) = bessel_j(v,z) + %i * bessel_y(v,z) and hankel_2(v,z) = bessel_j(v,z)  %i * bessel_y(v,z). If we try to get the Laplace transform of %h[v,sort](z), that is the Hankel function known to specint, we get: (%i2) specint(exp(s*t)*t^2*%h[1,1](t),t); Division by 0  an error. To debug this try debugmode(true); (%i3) specint(exp(s*t)*t^2*%h[1,2](t),t); Division by 0  an error. To debug this try debugmode(true); But both Laplace transforms Maxima can calculate: (%i7) factor(ratsimp(specint(exp(s*t)*t^2*(bessel_j(1,t)+%i*bessel_y(1,t)),t))); (%o7) (3*s*sqrt(s^2+1)*log((sqrt(s^2+1)+s)/(sqrt(s^2+1)s)) +3*%pi*s*sqrt(s^2+1)+2*s^42*s^24) /(%pi*(s^2+1)^3) (%i8) factor(ratsimp(specint(exp(s*t)*t^2*(bessel_j(1,t)%i*bessel_y(1,t)),t))); (%o8) (3*s*sqrt(s^2+1)*log((sqrt(s^2+1)+s)/(sqrt(s^2+1)s)) +3*%pi*s*sqrt(s^2+1)+2*s^42*s^24) /(%pi*(s^2+1)^3) The error for the hankel functions %h[v,sort](t) occurs, because the transformation to Bessel J and Bessel Y functions is implemented wrongly in the routine htjory. Dieter Kaiser  >Comment By: Dieter Kaiser (crategus) Date: 20090926 20:52 Message: Fixed in hypgeo.lisp revision 1.66. Closing this bug report. Dieter Kaiser  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867434&group_id=4933 
From: SourceForge.net <noreply@so...>  20090926 16:02:31

Bugs item #2867499, was opened at 20090926 18:02 Message generated for change (Tracker Item Submitted) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867499&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None Status: Open Resolution: None Priority: 5 Private: No Submitted By: Dieter Kaiser (crategus) Assigned to: Nobody/Anonymous (nobody) Summary: specint(exp(s*t)*CONST*t^2*bessel_y(1,t),t) wrong Initial Comment: Maxima does not take into a account a constant part of an integrand when doing the Laplace transform of bessel_y(v,a*t) where v is an integer order. The symbol CONST is missing in the following result: (%i5) ratsimp(specint(exp(s*t)*CONST*t^2*bessel_y(1,t),t)); (%o5) (3*s*log((sqrt(s^2+1)+s)/(sqrt(s^2+1)s))+sqrt(s^2+1)*(2*s^24)) /(%pi*sqrt(s^2+1)*(s^4+2*s^2+1)) The error is in the routine f2p105v2cond. The constant part is extracted from the variable l but not multiplied to the result of the Laplace transform. Dieter Kaiser  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867499&group_id=4933 
From: SourceForge.net <noreply@so...>  20090926 13:38:01

Bugs item #2866802, was opened at 20090925 20:23 Message generated for change (Settings changed) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2866802&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None >Status: Closed >Resolution: Fixed Priority: 5 Private: No Submitted By: Dieter Kaiser (crategus) Assigned to: Nobody/Anonymous (nobody) Summary: specint(exp(s*t)*t^(5/2)*bessel_j(1/2,sqrt(t))^2,t) wrong Initial Comment: We have the following known failure in rtest14.mac: ********************** Problem 57 *************** Input: specint(t^(5/2)*bessel_y(1/2,t^(1/2))^2*%e^(p*t),t) In $specint the expression with the bessel_y function is transformed to the square of the bessel_j function. So we get the following integrand: (1) t^(5/2)*(bessel_j(1/2,sqrt(t))^2 Furthermore, this expression is equivalent to: (2) 2/%pi*cos(sqrt(t))^2 Maxima can do this transformation: (%i17) t^(5/2)*bessel_j(1/2,sqrt(t))^2,besselexpand:true; (%o17) 2*cos(sqrt(t))^2*t^2/%pi The problem is that the integrands (1) and (2) give different Laplace transforms: First the result for bessel_j(1/2,sqrt(t))^2: (%i14) res1:factor(ratsimp(specint(exp(s*t)*t^(5/2)*bessel_j(1/2,sqrt(t))^2,t))); (%o14) %e^(1/s)*(8*s^3*%e^(1/s)18*s^2*%e^(1/s)+4*s*%e^(1/s) +15*sqrt(%pi)*%i*erf(%i/sqrt(s))*s^(5/2) 20*sqrt(%pi)*%i*erf(%i/sqrt(s))*s^(3/2) +4*sqrt(%pi)*%i*erf(%i/sqrt(s))*sqrt(s)) /(2*%pi*s^6) Next, the result for cos(sqrt(t))^2 (we use the flag besselexpand): (%i15) res2 : factor(ratsimp(specint(exp(s*t) * t^(5/2) * bessel_j(1/2, sqrt(t))^2, t))), besselexpand:true; (%o15) %e^(1/s)*(16*s^3*%e^(1/s)18*s^2*%e^(1/s)+4*s*%e^(1/s) +15*sqrt(%pi)*%i*erf(%i/sqrt(s))*s^(5/2) 20*sqrt(%pi)*%i*erf(%i/sqrt(s))*s^(3/2) +4*sqrt(%pi)*%i*erf(%i/sqrt(s))*sqrt(s)) /(4*%pi*s^6) The results differ by a factor 2 in most, but not in all terms. I had a long search for the bug and I have found the problem in the algorithm for the product of hypergeometric functions. Maxima does the following transformation for our case of two bessel_j(1/2,sqrt(t)) functions: bessel_j(1/2,sqrt(t))^2 > 2/%pi*2F3([0,1/2], [1/2,1/2,0], t) Next the hypergeometric function is reduced in two steps: 2F3([0,1/2], [1/2,1/2,0], t) > 1F2([1/2], [1/2,1/2], t) > 0F1([], [1/2], t) But, 0F1([],[1/2],t) represents cos(2*sqrt(t)) and not cos(sqrt(t))^2 as expected. Therefore, we get the Laplace transform of cos(2*sqrt(t)) and not of cos(sqrt(t))^2, when we use the hypergeometric algorithm. We can check this by doing the Laplace transform of cos(2*sqrt(t)) directly. A correct hypergeometric representation of cos(sqrt(t))^2 is 1/2*(0F1([],[1/2],t) + 1). The error is, that we do the following transformation for a parameter a=0: 2F3([a,1/2],[1/2,1/2,a],t) > 1F2([1/2], [1/2,1/2], t). I think this transformation is not valid for a=0, because the hypergeometric function 2F3 is not well defined for this case. The parameter a is zero for v+u=1, where v and u are the order of the two Bessel J functions involved in the Laplace transformation. For the square of bessel functions this is the case for v=1/2. Remark: There are a lot of more possibilities for wrong results. The example in rtest14.mac is not the simplest one. Dieter Kaiser  >Comment By: Dieter Kaiser (crategus) Date: 20090926 15:38 Message: The handling of the special case besse_j(1/2,t)^2 has been implemented. Now the Laplace transform of the example works as expected. Closing this bug report as fixed. Dieter Kaiser  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2866802&group_id=4933 
From: SourceForge.net <noreply@so...>  20090926 13:36:00

Bugs item #2867434, was opened at 20090926 15:35 Message generated for change (Tracker Item Submitted) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867434&group_id=4933 Please note that this message will contain a full copy of the comment thread, including the initial issue submission, for this request, not just the latest update. Category: Lisp Core  Integration Group: None Status: Open Resolution: None Priority: 5 Private: No Submitted By: Dieter Kaiser (crategus) Assigned to: Nobody/Anonymous (nobody) Summary: specint(exp(s*t)*t^2*%h[1,1](t),t) does not work Initial Comment: We know hankel_1(v,z) = bessel_j(v,z) + %i * bessel_y(v,z) and hankel_2(v,z) = bessel_j(v,z)  %i * bessel_y(v,z). If we try to get the Laplace transform of %h[v,sort](z), that is the Hankel function known to specint, we get: (%i2) specint(exp(s*t)*t^2*%h[1,1](t),t); Division by 0  an error. To debug this try debugmode(true); (%i3) specint(exp(s*t)*t^2*%h[1,2](t),t); Division by 0  an error. To debug this try debugmode(true); But both Laplace transforms Maxima can calculate: (%i7) factor(ratsimp(specint(exp(s*t)*t^2*(bessel_j(1,t)+%i*bessel_y(1,t)),t))); (%o7) (3*s*sqrt(s^2+1)*log((sqrt(s^2+1)+s)/(sqrt(s^2+1)s)) +3*%pi*s*sqrt(s^2+1)+2*s^42*s^24) /(%pi*(s^2+1)^3) (%i8) factor(ratsimp(specint(exp(s*t)*t^2*(bessel_j(1,t)%i*bessel_y(1,t)),t))); (%o8) (3*s*sqrt(s^2+1)*log((sqrt(s^2+1)+s)/(sqrt(s^2+1)s)) +3*%pi*s*sqrt(s^2+1)+2*s^42*s^24) /(%pi*(s^2+1)^3) The error for the hankel functions %h[v,sort](t) occurs, because the transformation to Bessel J and Bessel Y functions is implemented wrongly in the routine htjory. Dieter Kaiser  You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=2867434&group_id=4933 