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Maplesoft Document System Wed, 10 Jun 2026 19:00:44 GMT Wed, 10 Jun 2026 19:00:44 GMT The latest answers and comments added to the Question, Integral with GAMMA functions http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - answers and comments on Question, Integral with GAMMA functions http://www.mapleprimes.com/questions/143905-Integral-With-GAMMA-Functions By assuming http://www.mapleprimes.com/questions/143905-Integral-With-GAMMA-Functions?ref=Feed:MaplePrimes:Integral with GAMMA functions:Comments#answer143908 <p>It can be done as follows.</p> <p>&nbsp;</p> <form name="worksheet_form"> <table style="width: 576px;" align="center"> <tbody> <tr> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/2a5305f0fee6a1058e2153f8b1dbb572.gif" alt="A := beta^alpha*x^(alpha-1)*exp(-beta*x)/((1+x)*GAMMA(alpha));" width="488" height="23"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -19;" src="/view.aspx?sf=143908/454700/e165fdc3a24355cfecc37526c3ca3d23.gif" alt="beta^alpha*x^(alpha-1)*exp(-beta*x)/((1+x)*GAMMA(alpha))" width="100" height="55"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(1)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/cbb9eb0c4d3b18d3fb2215c5bbe16aa3.gif" alt="B := `assuming`([int(A, x = 0 .. infinity)], [alpha &gt; 0, alpha &lt; 1])" width="351" height="23"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -100;" src="/view.aspx?sf=143908/454700/8a547a7aca6f6eaf48ebf4d7af278a67.gif" alt="(-GAMMA(alpha-1)*sin(Pi*alpha)*alpha^2+GAMMA(alpha-1)*sin(Pi*alpha)*alpha*beta+GAMMA(alpha-1)*sin(Pi*alpha)*alpha-GAMMA(alpha-1)*sin(Pi*alpha)*beta-beta^(alpha-1)*exp(beta)*Pi*beta^2+alpha^3*beta^(alpha-1)*exp(beta)*GAMMA(-alpha-1, beta)*GAMMA(alpha-1)*beta^2*sin(Pi*alpha)-alpha*beta^(alpha-1)*exp(beta)*GAMMA(-alpha-1, beta)*GAMMA(alpha-1)*beta^2*sin(Pi*alpha)+Pi*beta^(alpha-1)*exp(beta)*csc(Pi*alpha)*beta^2*sin(Pi*alpha))/(beta*GAMMA(alpha)*sin(Pi*alpha))" width="546" height="126" align="middle"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(2)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -16;" src="/view.aspx?sf=143908/454700/c4b8d187f0d72eeeb21f0031e19b9d7b.gif" alt="eval(B, [alpha = 1/2, beta = 9])" width="197" height="42"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -23;" src="/view.aspx?sf=143908/454700/640256182217e03c000c6d3f6c3a010f.gif" alt="(1/9)*((17/2)*Pi^(1/2)+(27/4)*9^(1/2)*exp(9)*((4/3)*Pi^(1/2)*erfc(3)-(34/243)*9^(1/2)*exp(-9))*Pi^(1/2))/Pi^(1/2)" width="423" height="68"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(3)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/d326cda715c098398840dd1c1edd7cc7.gif" alt="evalf(%)" width="63" height="23"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/152196340b473d7c7f5395813ad36e12.gif" alt=".9518138382" width="92" height="23"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(4)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/3b8d5fdba016342c55602a26fe594128.gif" alt="``" width="11" height="23"></p> </td> </tr> </tbody> </table> <input type="hidden" name="sequence" value="1"> <input type="hidden" name="cmd" value="none"></form> <p>&nbsp;</p> <p><a href="/view.aspx?sf=143908/454700/int.mw">Download int.mw</a> See <a href="http://www.maplesoft.com/support/help/search.aspx?term=int,details">?int,details</a> , <a href="http://www.maplesoft.com/support/help/search.aspx?term=assume">?assume</a> , <a href="http://www.maplesoft.com/support/help/search.aspx?term=assuming">?assuming</a> , <a href="http://www.maplesoft.com/support/help/search.aspx?term=eval">?eval</a> for more info.</p> <p>PS. See the question I answer here <a href="/view.aspx?sf=143908/454711/screen25.02.13.docx">screen25.02.13.docx</a></p> <p>It can be done as follows.</p> <p>&nbsp;</p> <form name="worksheet_form"> <table style="width: 576px;" align="center"> <tbody> <tr> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/2a5305f0fee6a1058e2153f8b1dbb572.gif" alt="A := beta^alpha*x^(alpha-1)*exp(-beta*x)/((1+x)*GAMMA(alpha));" width="488" height="23"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -19;" src="/view.aspx?sf=143908/454700/e165fdc3a24355cfecc37526c3ca3d23.gif" alt="beta^alpha*x^(alpha-1)*exp(-beta*x)/((1+x)*GAMMA(alpha))" width="100" height="55"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(1)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/cbb9eb0c4d3b18d3fb2215c5bbe16aa3.gif" alt="B := `assuming`([int(A, x = 0 .. infinity)], [alpha &gt; 0, alpha &lt; 1])" width="351" height="23"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -100;" src="/view.aspx?sf=143908/454700/8a547a7aca6f6eaf48ebf4d7af278a67.gif" alt="(-GAMMA(alpha-1)*sin(Pi*alpha)*alpha^2+GAMMA(alpha-1)*sin(Pi*alpha)*alpha*beta+GAMMA(alpha-1)*sin(Pi*alpha)*alpha-GAMMA(alpha-1)*sin(Pi*alpha)*beta-beta^(alpha-1)*exp(beta)*Pi*beta^2+alpha^3*beta^(alpha-1)*exp(beta)*GAMMA(-alpha-1, beta)*GAMMA(alpha-1)*beta^2*sin(Pi*alpha)-alpha*beta^(alpha-1)*exp(beta)*GAMMA(-alpha-1, beta)*GAMMA(alpha-1)*beta^2*sin(Pi*alpha)+Pi*beta^(alpha-1)*exp(beta)*csc(Pi*alpha)*beta^2*sin(Pi*alpha))/(beta*GAMMA(alpha)*sin(Pi*alpha))" width="546" height="126" align="middle"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(2)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -16;" src="/view.aspx?sf=143908/454700/c4b8d187f0d72eeeb21f0031e19b9d7b.gif" alt="eval(B, [alpha = 1/2, beta = 9])" width="197" height="42"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -23;" src="/view.aspx?sf=143908/454700/640256182217e03c000c6d3f6c3a010f.gif" alt="(1/9)*((17/2)*Pi^(1/2)+(27/4)*9^(1/2)*exp(9)*((4/3)*Pi^(1/2)*erfc(3)-(34/243)*9^(1/2)*exp(-9))*Pi^(1/2))/Pi^(1/2)" width="423" height="68"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(3)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/d326cda715c098398840dd1c1edd7cc7.gif" alt="evalf(%)" width="63" height="23"></p> <table> <tbody> <tr valign="baseline"> <td> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/152196340b473d7c7f5395813ad36e12.gif" alt=".9518138382" width="92" height="23"></p> </td> <td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right">(4)</td> </tr> </tbody> </table> <p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"><img style="vertical-align: -6;" src="/view.aspx?sf=143908/454700/3b8d5fdba016342c55602a26fe594128.gif" alt="``" width="11" height="23"></p> </td> </tr> </tbody> </table> <input type="hidden" name="sequence" value="1"> <input type="hidden" name="cmd" value="none"></form> <p>&nbsp;</p> <p><a href="/view.aspx?sf=143908/454700/int.mw">Download int.mw</a> See <a href="http://www.maplesoft.com/support/help/search.aspx?term=int,details">?int,details</a> , <a href="http://www.maplesoft.com/support/help/search.aspx?term=assume">?assume</a> , <a href="http://www.maplesoft.com/support/help/search.aspx?term=assuming">?assuming</a> , <a href="http://www.maplesoft.com/support/help/search.aspx?term=eval">?eval</a> for more info.</p> <p>PS. See the question I answer here <a href="/view.aspx?sf=143908/454711/screen25.02.13.docx">screen25.02.13.docx</a></p> 143908 Mon, 25 Feb 2013 23:23:50 Z Markiyan Hirnyk Markiyan Hirnyk Integer http://www.mapleprimes.com/questions/143905-Integral-With-GAMMA-Functions?ref=Feed:MaplePrimes:Integral with GAMMA functions:Comments#comment143926 <p>How do I also the condition that alpha is an integer (positive)?</p> <p>How do I also the condition that alpha is an integer (positive)?</p> 143926 Tue, 26 Feb 2013 14:45:22 Z manueleleonelli manueleleonelli Assuming beta too http://www.mapleprimes.com/questions/143905-Integral-With-GAMMA-Functions?ref=Feed:MaplePrimes:Integral with GAMMA functions:Comments#comment143933 <p><a href="http://www.mapleprimes.com/questions/143905-Integral-With-GAMMA-Functions#comment143926">@manueleleonelli</a>&nbsp;</p> <p>&gt;int(A, x = 0 .. infinity) assuming alpha&gt;0, Re(beta) &gt; 0;</p> <p><img src="data:image/png;base64,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" alt=""></p> <p>&gt; evalf(eval(%, [alpha = 8, beta = 4+I]));<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.3566945739 + 0.05485185483 I</p> <p>Edit. My code: Re(beta) instead of beta.<br><br></p> <p>&nbsp;</p> <p><br><br></p> <p><a href="http://www.mapleprimes.com/questions/143905-Integral-With-GAMMA-Functions#comment143926">@manueleleonelli</a>&nbsp;</p> <p>&gt;int(A, x = 0 .. infinity) assuming alpha&gt;0, Re(beta) &gt; 0;</p> <p><img src="data:image/png;base64,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" alt=""></p> <p>&gt; evalf(eval(%, [alpha = 8, beta = 4+I]));<br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 0.3566945739 + 0.05485185483 I</p> <p>Edit. My code: Re(beta) instead of beta.<br><br></p> <p>&nbsp;</p> <p><br><br></p> 143933 Tue, 26 Feb 2013 17:51:14 Z Markiyan Hirnyk Markiyan Hirnyk