maxima-bugs

 [Maxima-bugs] [ maxima-Bugs-3387042 ] integrate(x exp(-x^2)) : strange jump From: SourceForge.net - 2011-08-05 22:04:04 ```Bugs item #3387042, was opened at 2011-08-06 02:04 Message generated for change (Tracker Item Submitted) made by kpf You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&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: Petr Kartsev (kpf) Assigned to: Nobody/Anonymous (nobody) Summary: integrate(x exp(-x^2)) : strange jump Initial Comment: (%i1) integrate( t*exp(-(t-a)^2), t, -inf, x); (%o1) -(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a*x)/(2*abs(x-a))+(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a^2)/(2*abs(x-a))-gamma_incomplete(1,x^2-2*a*x+a^2)/2 (1st part on screenshot) why is it so complex? when plotting this integral, it has strange jump at point x=1 (and even becomes negative) while the integrand is smooth and mostly always positive) (%i2) plot2d([x*exp(-(x-1)^2), integrate(t*exp(-(t-1)^2), t, -inf,x)], [x,-5,5])\$ (2nd part on screenshot) However, if I simply add and subtract unity, the actual result is recovered: (%i3) integrate( (t-1)*exp(-(t-1)^2), t, -inf, x)+ integrate( exp(-(t-1)^2), t, -inf, x); (%o3) (sqrt(%pi)*erf(x-1))/2-%e^(-x^2+2*x-1)/2+sqrt(%pi)/2 (3rd part on screenshot) Looks like additional signum or abs takes place somewhere? Sorry for not giving solution, I am not that skillful in lisp... Thanks for good system, anyway! :) ------------------------------------------------------------- Maxima version: 5.24.0 Maxima build date: 10:35 8/1/2011 Host type: x86_64-pc-linux-gnu Lisp implementation type: SBCL Lisp implementation version: 1.0.19-gentoo ------------------------------------------------------------- PS. Just for information, if this can help, run_testsuite() gives 1 error. May be just my installation is wrong? Running tests in rtest16: ********************** Problem 386 *************** Input: closeto(zeta(%i + 3) - (1.10721440843141 - .1482908671781754 %i), 1.e-15) Result: 3.3157171357748244e-9 This differed from the expected result: true ------------------------------------------- ---------------------------------------------------------------------- You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&group_id=4933 ```
 [Maxima-bugs] [ maxima-Bugs-3387042 ] integrate(x exp(-x^2)) : strange jump From: SourceForge.net - 2011-08-06 01:25:06 ```Bugs item #3387042, was opened at 2011-08-05 17:04 Message generated for change (Comment added) made by willisbl You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&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: Petr Kartsev (kpf) Assigned to: Nobody/Anonymous (nobody) Summary: integrate(x exp(-x^2)) : strange jump Initial Comment: (%i1) integrate( t*exp(-(t-a)^2), t, -inf, x); (%o1) -(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a*x)/(2*abs(x-a))+(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a^2)/(2*abs(x-a))-gamma_incomplete(1,x^2-2*a*x+a^2)/2 (1st part on screenshot) why is it so complex? when plotting this integral, it has strange jump at point x=1 (and even becomes negative) while the integrand is smooth and mostly always positive) (%i2) plot2d([x*exp(-(x-1)^2), integrate(t*exp(-(t-1)^2), t, -inf,x)], [x,-5,5])\$ (2nd part on screenshot) However, if I simply add and subtract unity, the actual result is recovered: (%i3) integrate( (t-1)*exp(-(t-1)^2), t, -inf, x)+ integrate( exp(-(t-1)^2), t, -inf, x); (%o3) (sqrt(%pi)*erf(x-1))/2-%e^(-x^2+2*x-1)/2+sqrt(%pi)/2 (3rd part on screenshot) Looks like additional signum or abs takes place somewhere? Sorry for not giving solution, I am not that skillful in lisp... Thanks for good system, anyway! :) ------------------------------------------------------------- Maxima version: 5.24.0 Maxima build date: 10:35 8/1/2011 Host type: x86_64-pc-linux-gnu Lisp implementation type: SBCL Lisp implementation version: 1.0.19-gentoo ------------------------------------------------------------- PS. Just for information, if this can help, run_testsuite() gives 1 error. May be just my installation is wrong? Running tests in rtest16: ********************** Problem 386 *************** Input: closeto(zeta(%i + 3) - (1.10721440843141 - .1482908671781754 %i), 1.e-15) Result: 3.3157171357748244e-9 This differed from the expected result: true ------------------------------------------- ---------------------------------------------------------------------- >Comment By: Barton Willis (willisbl) Date: 2011-08-05 20:25 Message: First, you should check your initialization file--maybe you have set some option variables to non default values ; second try this: (%i23) e1 : integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?"neg; (%o23) -(gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))+gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2 (%i24) e2 : integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?"pos; (%o24) (gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))-gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) (%i25) plot2d(if x < 1 then ''e1 else ''e2,[x,-10,10]); The graph appears to be continuous. ---------------------------------------------------------------------- You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&group_id=4933 ```
 [Maxima-bugs] [ maxima-Bugs-3387042 ] integrate(x exp(-x^2)) : strange jump From: SourceForge.net - 2011-08-11 22:30:46 ```Bugs item #3387042, was opened at 2011-08-06 02:04 Message generated for change (Comment added) made by kpf You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&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: Petr Kartsev (kpf) Assigned to: Nobody/Anonymous (nobody) Summary: integrate(x exp(-x^2)) : strange jump Initial Comment: (%i1) integrate( t*exp(-(t-a)^2), t, -inf, x); (%o1) -(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a*x)/(2*abs(x-a))+(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a^2)/(2*abs(x-a))-gamma_incomplete(1,x^2-2*a*x+a^2)/2 (1st part on screenshot) why is it so complex? when plotting this integral, it has strange jump at point x=1 (and even becomes negative) while the integrand is smooth and mostly always positive) (%i2) plot2d([x*exp(-(x-1)^2), integrate(t*exp(-(t-1)^2), t, -inf,x)], [x,-5,5])\$ (2nd part on screenshot) However, if I simply add and subtract unity, the actual result is recovered: (%i3) integrate( (t-1)*exp(-(t-1)^2), t, -inf, x)+ integrate( exp(-(t-1)^2), t, -inf, x); (%o3) (sqrt(%pi)*erf(x-1))/2-%e^(-x^2+2*x-1)/2+sqrt(%pi)/2 (3rd part on screenshot) Looks like additional signum or abs takes place somewhere? Sorry for not giving solution, I am not that skillful in lisp... Thanks for good system, anyway! :) ------------------------------------------------------------- Maxima version: 5.24.0 Maxima build date: 10:35 8/1/2011 Host type: x86_64-pc-linux-gnu Lisp implementation type: SBCL Lisp implementation version: 1.0.19-gentoo ------------------------------------------------------------- PS. Just for information, if this can help, run_testsuite() gives 1 error. May be just my installation is wrong? Running tests in rtest16: ********************** Problem 386 *************** Input: closeto(zeta(%i + 3) - (1.10721440843141 - .1482908671781754 %i), 1.e-15) Result: 3.3157171357748244e-9 This differed from the expected result: true ------------------------------------------- ---------------------------------------------------------------------- >Comment By: Petr Kartsev (kpf) Date: 2011-08-12 02:30 Message: willisbl, thanks for answer. I upgraded to version 5.25 (appeared in Gentoo Linux repository ) and now the behavior is exactly as you posted! However, my question transforms to the following, may be I should file it as dedicated bug report: Can you remove the question "positive/negative/zero" in this situation, as we can see the result is the same (already known erf+exp) for all cases ? This can be possible, in my opinion, since this question is already removed by simply shifting the variable: (%i6) e1(x): integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?" pos; (%o6) (gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))-gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) but (%i8) e1: integrate( (t+1)*exp(-t^2), t,minf,x-1); (%o8) (%e^(-x^2-1)*(sqrt(%pi)*%e^(x^2+1)*erf(x-1)-%e^(2*x)))/2+sqrt(%pi)/2 (My reason is that in my work I differentiate this function and then do some numerical actions for programmed solution with external parameters. Such automatic analysis would get in trouble if it needs to answer positive/negative) Maybe gamma_incomplete is too universal to let simple erf appear in the formula, but erf is sometimes better, so can some special case be implemented? Sorry if I see something wrong. Thanks for attention and for this good tool! Sincerely, Petr F. Kartsev ---------------------------------------------------------------------- Comment By: Barton Willis (willisbl) Date: 2011-08-06 05:25 Message: First, you should check your initialization file--maybe you have set some option variables to non default values ; second try this: (%i23) e1 : integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?"neg; (%o23) -(gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))+gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2 (%i24) e2 : integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?"pos; (%o24) (gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))-gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) (%i25) plot2d(if x < 1 then ''e1 else ''e2,[x,-10,10]); The graph appears to be continuous. ---------------------------------------------------------------------- You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&group_id=4933 ```
 [Maxima-bugs] [ maxima-Bugs-3387042 ] integrate(x exp(-x^2)) : strange jump From: SourceForge.net - 2011-08-11 23:51:50 ```Bugs item #3387042, was opened at 2011-08-06 00:04 Message generated for change (Comment added) made by crategus You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&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: Pending >Resolution: Works For Me Priority: 5 Private: No Submitted By: Petr Kartsev (kpf) Assigned to: Nobody/Anonymous (nobody) Summary: integrate(x exp(-x^2)) : strange jump Initial Comment: (%i1) integrate( t*exp(-(t-a)^2), t, -inf, x); (%o1) -(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a*x)/(2*abs(x-a))+(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a^2)/(2*abs(x-a))-gamma_incomplete(1,x^2-2*a*x+a^2)/2 (1st part on screenshot) why is it so complex? when plotting this integral, it has strange jump at point x=1 (and even becomes negative) while the integrand is smooth and mostly always positive) (%i2) plot2d([x*exp(-(x-1)^2), integrate(t*exp(-(t-1)^2), t, -inf,x)], [x,-5,5])\$ (2nd part on screenshot) However, if I simply add and subtract unity, the actual result is recovered: (%i3) integrate( (t-1)*exp(-(t-1)^2), t, -inf, x)+ integrate( exp(-(t-1)^2), t, -inf, x); (%o3) (sqrt(%pi)*erf(x-1))/2-%e^(-x^2+2*x-1)/2+sqrt(%pi)/2 (3rd part on screenshot) Looks like additional signum or abs takes place somewhere? Sorry for not giving solution, I am not that skillful in lisp... Thanks for good system, anyway! :) ------------------------------------------------------------- Maxima version: 5.24.0 Maxima build date: 10:35 8/1/2011 Host type: x86_64-pc-linux-gnu Lisp implementation type: SBCL Lisp implementation version: 1.0.19-gentoo ------------------------------------------------------------- PS. Just for information, if this can help, run_testsuite() gives 1 error. May be just my installation is wrong? Running tests in rtest16: ********************** Problem 386 *************** Input: closeto(zeta(%i + 3) - (1.10721440843141 - .1482908671781754 %i), 1.e-15) Result: 3.3157171357748244e-9 This differed from the expected result: true ------------------------------------------- ---------------------------------------------------------------------- >Comment By: Dieter Kaiser (crategus) Date: 2011-08-12 01:51 Message: One way to avoid questions is to give Maxima facts which are stored in the assume database, e.g. (%i1) assume(x>1); (%o1) [x > 1] (%i2) res:integrate( t*exp(-(t-1)^2), t,minf,x); (%o2) gamma_incomplete(1/2,x^2-2*x+1)*x/(2*(1-x)) -gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x)) -gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) The result can be transformed to a representation in terms of the erfc function with the option variable gamma_expand: (%i3) res, gamma_expand:true; (%o3) sqrt(%pi)*x*erfc(sqrt(x^2-2*x+1))/(2*(1-x)) -sqrt(%pi)*erfc(sqrt(x^2-2*x+1))/(2*(1-x))-%e^(-x^2+2*x-1)/2+sqrt(%pi) In addition it is possible to transform to the erf function when setting the option variable erf_representation: (%i4) res,gamma_expand:true, erf_representation:erf; (%o4) sqrt(%pi)*x*(1-erf(sqrt(x^2-2*x+1)))/(2*(1-x)) -sqrt(%pi)*(1-erf(sqrt(x^2-2*x+1)))/(2*(1-x))-%e^(-x^2+2*x-1)/2 +sqrt(%pi) At this point I think we do not have a bug. I suggest to close this bug report as "works for me". Setting the status to pending. Dieter Kaiser ---------------------------------------------------------------------- Comment By: Petr Kartsev (kpf) Date: 2011-08-12 00:30 Message: willisbl, thanks for answer. I upgraded to version 5.25 (appeared in Gentoo Linux repository ) and now the behavior is exactly as you posted! However, my question transforms to the following, may be I should file it as dedicated bug report: Can you remove the question "positive/negative/zero" in this situation, as we can see the result is the same (already known erf+exp) for all cases ? This can be possible, in my opinion, since this question is already removed by simply shifting the variable: (%i6) e1(x): integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?" pos; (%o6) (gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))-gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) but (%i8) e1: integrate( (t+1)*exp(-t^2), t,minf,x-1); (%o8) (%e^(-x^2-1)*(sqrt(%pi)*%e^(x^2+1)*erf(x-1)-%e^(2*x)))/2+sqrt(%pi)/2 (My reason is that in my work I differentiate this function and then do some numerical actions for programmed solution with external parameters. Such automatic analysis would get in trouble if it needs to answer positive/negative) Maybe gamma_incomplete is too universal to let simple erf appear in the formula, but erf is sometimes better, so can some special case be implemented? Sorry if I see something wrong. Thanks for attention and for this good tool! Sincerely, Petr F. Kartsev ---------------------------------------------------------------------- Comment By: Barton Willis (willisbl) Date: 2011-08-06 03:25 Message: First, you should check your initialization file--maybe you have set some option variables to non default values ; second try this: (%i23) e1 : integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?"neg; (%o23) -(gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))+gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2 (%i24) e2 : integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?"pos; (%o24) (gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))-gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) (%i25) plot2d(if x < 1 then ''e1 else ''e2,[x,-10,10]); The graph appears to be continuous. ---------------------------------------------------------------------- You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&group_id=4933 ```
 [Maxima-bugs] [ maxima-Bugs-3387042 ] integrate(x exp(-x^2)) : strange jump From: SourceForge.net - 2012-05-03 20:45:37 ```Bugs item #3387042, was opened at 2011-08-05 15:04 Message generated for change (Comment added) made by rtoy You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&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: Works For Me Priority: 5 Private: No Submitted By: Petr Kartsev (kpf) Assigned to: Nobody/Anonymous (nobody) Summary: integrate(x exp(-x^2)) : strange jump Initial Comment: (%i1) integrate( t*exp(-(t-a)^2), t, -inf, x); (%o1) -(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a*x)/(2*abs(x-a))+(gamma_incomplete(1/2,x^2-2*a*x+a^2)*a^2)/(2*abs(x-a))-gamma_incomplete(1,x^2-2*a*x+a^2)/2 (1st part on screenshot) why is it so complex? when plotting this integral, it has strange jump at point x=1 (and even becomes negative) while the integrand is smooth and mostly always positive) (%i2) plot2d([x*exp(-(x-1)^2), integrate(t*exp(-(t-1)^2), t, -inf,x)], [x,-5,5])\$ (2nd part on screenshot) However, if I simply add and subtract unity, the actual result is recovered: (%i3) integrate( (t-1)*exp(-(t-1)^2), t, -inf, x)+ integrate( exp(-(t-1)^2), t, -inf, x); (%o3) (sqrt(%pi)*erf(x-1))/2-%e^(-x^2+2*x-1)/2+sqrt(%pi)/2 (3rd part on screenshot) Looks like additional signum or abs takes place somewhere? Sorry for not giving solution, I am not that skillful in lisp... Thanks for good system, anyway! :) ------------------------------------------------------------- Maxima version: 5.24.0 Maxima build date: 10:35 8/1/2011 Host type: x86_64-pc-linux-gnu Lisp implementation type: SBCL Lisp implementation version: 1.0.19-gentoo ------------------------------------------------------------- PS. Just for information, if this can help, run_testsuite() gives 1 error. May be just my installation is wrong? Running tests in rtest16: ********************** Problem 386 *************** Input: closeto(zeta(%i + 3) - (1.10721440843141 - .1482908671781754 %i), 1.e-15) Result: 3.3157171357748244e-9 This differed from the expected result: true ------------------------------------------- ---------------------------------------------------------------------- >Comment By: Raymond Toy (rtoy) Date: 2012-05-03 13:45 Message: Closing. ---------------------------------------------------------------------- Comment By: Dieter Kaiser (crategus) Date: 2011-08-11 16:51 Message: One way to avoid questions is to give Maxima facts which are stored in the assume database, e.g. (%i1) assume(x>1); (%o1) [x > 1] (%i2) res:integrate( t*exp(-(t-1)^2), t,minf,x); (%o2) gamma_incomplete(1/2,x^2-2*x+1)*x/(2*(1-x)) -gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x)) -gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) The result can be transformed to a representation in terms of the erfc function with the option variable gamma_expand: (%i3) res, gamma_expand:true; (%o3) sqrt(%pi)*x*erfc(sqrt(x^2-2*x+1))/(2*(1-x)) -sqrt(%pi)*erfc(sqrt(x^2-2*x+1))/(2*(1-x))-%e^(-x^2+2*x-1)/2+sqrt(%pi) In addition it is possible to transform to the erf function when setting the option variable erf_representation: (%i4) res,gamma_expand:true, erf_representation:erf; (%o4) sqrt(%pi)*x*(1-erf(sqrt(x^2-2*x+1)))/(2*(1-x)) -sqrt(%pi)*(1-erf(sqrt(x^2-2*x+1)))/(2*(1-x))-%e^(-x^2+2*x-1)/2 +sqrt(%pi) At this point I think we do not have a bug. I suggest to close this bug report as "works for me". Setting the status to pending. Dieter Kaiser ---------------------------------------------------------------------- Comment By: Petr Kartsev (kpf) Date: 2011-08-11 15:30 Message: willisbl, thanks for answer. I upgraded to version 5.25 (appeared in Gentoo Linux repository ) and now the behavior is exactly as you posted! However, my question transforms to the following, may be I should file it as dedicated bug report: Can you remove the question "positive/negative/zero" in this situation, as we can see the result is the same (already known erf+exp) for all cases ? This can be possible, in my opinion, since this question is already removed by simply shifting the variable: (%i6) e1(x): integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?" pos; (%o6) (gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))-gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) but (%i8) e1: integrate( (t+1)*exp(-t^2), t,minf,x-1); (%o8) (%e^(-x^2-1)*(sqrt(%pi)*%e^(x^2+1)*erf(x-1)-%e^(2*x)))/2+sqrt(%pi)/2 (My reason is that in my work I differentiate this function and then do some numerical actions for programmed solution with external parameters. Such automatic analysis would get in trouble if it needs to answer positive/negative) Maybe gamma_incomplete is too universal to let simple erf appear in the formula, but erf is sometimes better, so can some special case be implemented? Sorry if I see something wrong. Thanks for attention and for this good tool! Sincerely, Petr F. Kartsev ---------------------------------------------------------------------- Comment By: Barton Willis (willisbl) Date: 2011-08-05 18:25 Message: First, you should check your initialization file--maybe you have set some option variables to non default values ; second try this: (%i23) e1 : integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?"neg; (%o23) -(gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))+gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2 (%i24) e2 : integrate( t*exp(-(t-1)^2), t,minf,x); "Is "x-1" positive, negative, or zero?"pos; (%o24) (gamma_incomplete(1/2,x^2-2*x+1)*x)/(2*(1-x))-gamma_incomplete(1/2,x^2-2*x+1)/(2*(1-x))-gamma_incomplete(1,x^2-2*x+1)/2+sqrt(%pi) (%i25) plot2d(if x < 1 then ''e1 else ''e2,[x,-10,10]); The graph appears to be continuous. ---------------------------------------------------------------------- You can respond by visiting: https://sourceforge.net/tracker/?func=detail&atid=104933&aid=3387042&group_id=4933 ```