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en-us2015 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemWed, 01 Apr 2015 01:09:54 GMTWed, 01 Apr 2015 01:09:54 GMTThe most recent questions and posts on MaplePrimes tagged with asymptotichttp://www.mapleprimes.com/images/mapleprimeswhite.jpgMaplePrimes - Questions and Posts tagged with asymptotic
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Euler-Maclaurin Summation
http://www.mapleprimes.com/posts/200229-EulerMaclaurin-Summation?ref=Feed:MaplePrimes:Tagged With asymptotic
<p>Greetings to all.</p>
<p>I would like to share a brief observation concerning my experiences with the Euler-Maclaurin summation routine in <strong>Maple 17 (X86 64 LINUX)</strong>. The following <a href="http://math.stackexchange.com/questions/740088/use-the-euler-maclaurin-summation-formula-to-estimate-a-summation">Math StackExchange Link</a> shows how to compute a certain Euler-MacLaurin type asymptotic expansion using highly unorthodox divergent series summation techniques. The result that was obtained matches the output from <strong>eulermac</strong> which is definitely good to know. What follows is the output from said routine.</p>
<pre>> eulermac(1/(1+k/n),k=0..n,18);
1 929569 3202291 691 O(1)
O(- ---) - ----------- + ----------- - --------- + 1/1048576 ----
19 15 17 11 19
n 2097152 n 1048576 n 32768 n n
n
/
174611 5461 31 | 1 17 1
- -------- + --------- + ------- + | ------- dk - ------- + ------
19 13 9 | 1 + k/n 7 5
6600 n 65536 n 4096 n / 4096 n 256 n
0
1 1
- ------ + ---- + 3/4
3 16 n
128 n
</pre>
<p>While I realize that this is good enough for most purposes I have two minor issues.</p>
<ul>
<li>One could certainly evaluate the integral without leaving it to the user to force evaluation with the <strong>AllSolutions</strong> option. One can and should make use of what is known about n and k. In particular one can check whether there are singularities on the integration path because we know the range of <em>k/n</em>.</li>
<li>Why are there two order terms for the order of the remainder term? There should be at most one and a coefficient times an <em>O(1)</em> term makes little sense as the coefficient would be absorbed.</li>
</ul>
<p>You might want to fix these so that the output looks a bit more professional which does enter into play when potential future users decide on what CAS to commit to. Other than that it is a very useful routine even for certain harmonic sum computations where one can use Euler-Maclaurin to verify results.</p>
<p>Best regards,</p>
<p>Marko Riedel</p><p>Greetings to all.</p>
<p>I would like to share a brief observation concerning my experiences with the Euler-Maclaurin summation routine in <strong>Maple 17 (X86 64 LINUX)</strong>. The following <a href="http://math.stackexchange.com/questions/740088/use-the-euler-maclaurin-summation-formula-to-estimate-a-summation">Math StackExchange Link</a> shows how to compute a certain Euler-MacLaurin type asymptotic expansion using highly unorthodox divergent series summation techniques. The result that was obtained matches the output from <strong>eulermac</strong> which is definitely good to know. What follows is the output from said routine.</p>
<pre>> eulermac(1/(1+k/n),k=0..n,18);
1 929569 3202291 691 O(1)
O(- ---) - ----------- + ----------- - --------- + 1/1048576 ----
19 15 17 11 19
n 2097152 n 1048576 n 32768 n n
n
/
174611 5461 31 | 1 17 1
- -------- + --------- + ------- + | ------- dk - ------- + ------
19 13 9 | 1 + k/n 7 5
6600 n 65536 n 4096 n / 4096 n 256 n
0
1 1
- ------ + ---- + 3/4
3 16 n
128 n
</pre>
<p>While I realize that this is good enough for most purposes I have two minor issues.</p>
<ul>
<li>One could certainly evaluate the integral without leaving it to the user to force evaluation with the <strong>AllSolutions</strong> option. One can and should make use of what is known about n and k. In particular one can check whether there are singularities on the integration path because we know the range of <em>k/n</em>.</li>
<li>Why are there two order terms for the order of the remainder term? There should be at most one and a coefficient times an <em>O(1)</em> term makes little sense as the coefficient would be absorbed.</li>
</ul>
<p>You might want to fix these so that the output looks a bit more professional which does enter into play when potential future users decide on what CAS to commit to. Other than that it is a very useful routine even for certain harmonic sum computations where one can use Euler-Maclaurin to verify results.</p>
<p>Best regards,</p>
<p>Marko Riedel</p>200229Sun, 06 Apr 2014 21:18:57 Zmriedelmriedelsimple asymptotic expansion
http://www.mapleprimes.com/questions/144127-Simple-Asymptotic-Expansion?ref=Feed:MaplePrimes:Tagged With asymptotic
<p>Dear friends,</p>
<p>I wonder how I would go about calculating the asymptotic expansion of</p>
<pre>sum(5^j/j, j=1..m+1)?</pre>
<p>The motivation for this calculation can be found <a href="http://math.stackexchange.com/questions/318247/tn-t-fracn5-frac-n-log-n-solving">here</a>. The correct answer is</p>
<pre>5/4 5^(m+1)/(m+1).</pre>
<p>The classic asympt and the one from multiseries both fail on this one.</p>
<p>Thanks,</p>
<p>Marko Riedel</p><p>Dear friends,</p>
<p>I wonder how I would go about calculating the asymptotic expansion of</p>
<pre>sum(5^j/j, j=1..m+1)?</pre>
<p>The motivation for this calculation can be found <a href="http://math.stackexchange.com/questions/318247/tn-t-fracn5-frac-n-log-n-solving">here</a>. The correct answer is</p>
<pre>5/4 5^(m+1)/(m+1).</pre>
<p>The classic asympt and the one from multiseries both fail on this one.</p>
<p>Thanks,</p>
<p>Marko Riedel</p>144127Sun, 03 Mar 2013 02:42:50 Zmriedelmriedelhow to plot Sigma(R30)?
http://www.mapleprimes.com/questions/124946-How-To-Plot-SigmaR30?ref=Feed:MaplePrimes:Tagged With asymptotic
<p>Hi,</p>
<p>I am trying to plot this function Sigma(R30) but I get failed to do so. Any one would like to try to help me out?</p>
<p>The attached maple sheet contains the asymptotic solution of the huge equation in .txt file. </p>
<p>thx.</p>
<p> <a href="/ViewTemp.ashx?f=68690_1313884198/20110819_doodles2.mws">20110819_doodles2.mws</a></p>
<p><a href="/ViewTemp.ashx?f=68690_1313884198/20110818_section4-5a.txt">20110818_section4-5a.txt</a></p><p>Hi,</p>
<p>I am trying to plot this function Sigma(R30) but I get failed to do so. Any one would like to try to help me out?</p>
<p>The attached maple sheet contains the asymptotic solution of the huge equation in .txt file. </p>
<p>thx.</p>
<p> <a href="/view.aspx?sf=124946/419514/20110819_doodles2.mws">20110819_doodles2.mws</a></p>
<p><a href="/view.aspx?sf=124946/419514/20110818_section4-5a.txt">20110818_section4-5a.txt</a></p>124946Sun, 21 Aug 2011 03:57:05 Zkh2nkh2nOnce again an Asymptotic series problem
http://www.mapleprimes.com/questions/123502-Once-Again-An-Asymptotic-Series-Problem?ref=Feed:MaplePrimes:Tagged With asymptotic
<p>I am sorry for bothering you all with the asymptotic again and again. Actually I am unable to find a magic way to evaluate an asymptotic expension. </p>
<p><a href="/ViewTemp.ashx?f=68690_1309598042/R3Infintyplots_S0=.mws">R3Infintyplots_S0=.mws</a></p><p>I am sorry for bothering you all with the asymptotic again and again. Actually I am unable to find a magic way to evaluate an asymptotic expension. </p>
<p><a href="/view.aspx?sf=123502/416499/R3Infintyplots_S0=.mws">R3Infintyplots_S0=.mws</a></p>123502Sat, 02 Jul 2011 13:16:56 Zkh2nkh2nHankel function asymptotic form
http://www.mapleprimes.com/questions/120722-Hankel-Function-Asymptotic-Form?ref=Feed:MaplePrimes:Tagged With asymptotic
<p>In Arfken(Mathematical methods for physicists,5-th edition,page 483),the asymptotic form of the Hankel function is approximated as</p>
<p>H1(t,s)=</p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=b79f7cc2044db45376811c9571368560.gif" alt="sqrt(2/(Pi*s))*exp(I*(s-t*(Pi/2)-Pi/4))"></p>
<p>Is there any simple/direct way in Maple(using HankelH1(),or otherwise) to achieve this?I don't want to assign numerical values to t or s.</p><p>In Arfken(Mathematical methods for physicists,5-th edition,page 483),the asymptotic form of the Hankel function is approximated as</p>
<p>H1(t,s)=</p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=b79f7cc2044db45376811c9571368560.gif" alt="sqrt(2/(Pi*s))*exp(I*(s-t*(Pi/2)-Pi/4))"></p>
<p>Is there any simple/direct way in Maple(using HankelH1(),or otherwise) to achieve this?I don't want to assign numerical values to t or s.</p>120722Mon, 30 May 2011 19:44:04 Zhermitianhermitianasymptotics of Lambert W
http://www.mapleprimes.com/questions/120720-Asymptotics-Of-Lambert-W?ref=Feed:MaplePrimes:Tagged With asymptotic
<p>I'm using Maple 15. It seems to me this worked in some previous version...</p>
<p>Consider the Lambert W function, y=LambertW(0,x) ... I want Maple to tell me the asymptotics for it,</p>
<p>something like this:</p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=d0b124c3849a46834c98a3ee3dc8a04b.gif" alt="log(x)-log(log(x))+log(log(x))/log(x)"></p>
<p>But I don't get that now. Is my memory faulty that I got it in the past?</p>
<p>Maple 15:</p>
<p> <strong>asympt(LambertW(0,x),x);</strong></p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=1a2407e812dccbc02b0a1b0e474e2553.gif" alt="asympt(LambertW(0,x),x)"></p>
<p>not very useful...</p>
<pre><strong>series(LambertW(0,x),x=infinity);</strong></pre>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=72c4af6ec1f760ec1c1fcd52081c6de7.gif" alt="LambertW(0,x)"></p>
<pre><br><strong>with(MultiSeries):<br>series(LambertW(0,x),x=infinity);</strong></pre>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=72c4af6ec1f760ec1c1fcd52081c6de7.gif" alt="LambertW(0,x)"> </p>
<p>I'm using Maple 15. It seems to me this worked in some previous version...</p>
<p>Consider the Lambert W function, y=LambertW(0,x) ... I want Maple to tell me the asymptotics for it,</p>
<p>something like this:</p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=d0b124c3849a46834c98a3ee3dc8a04b.gif" alt="log(x)-log(log(x))+log(log(x))/log(x)"></p>
<p>But I don't get that now. Is my memory faulty that I got it in the past?</p>
<p>Maple 15:</p>
<p> <strong>asympt(LambertW(0,x),x);</strong></p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=1a2407e812dccbc02b0a1b0e474e2553.gif" alt="asympt(LambertW(0,x),x)"></p>
<p>not very useful...</p>
<pre><strong>series(LambertW(0,x),x=infinity);</strong></pre>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=72c4af6ec1f760ec1c1fcd52081c6de7.gif" alt="LambertW(0,x)"></p>
<pre><br><strong>with(MultiSeries):<br>series(LambertW(0,x),x=infinity);</strong></pre>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=72c4af6ec1f760ec1c1fcd52081c6de7.gif" alt="LambertW(0,x)"> </p>
<!--break-->
<p>---</p>
<p>G A Edgar</p>120720Mon, 30 May 2011 17:57:34 Zedgaredgarwhat is the asymptotic constant of
http://www.mapleprimes.com/questions/102406-What-Is-The-Asymptotic-Constant-Of?ref=Feed:MaplePrimes:Tagged With asymptotic
<p> </p>
<p>I am interested in finding the asymptofic constant (Big O(1/n^(2m+2)) for the following expansion</p>
<p> </p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=acd54f9d5aec18c25d2b64bb3b8072e6.gif" alt="series((1+1/n)^((1/2)/(sum(1/((2*k+1)*(2*n+1)^(2*k+1)), k = 0 .. m))), n = infinity, 10)"></p>
<p> </p>
<p>Upon using the preceding command in the maple i get</p>
<p> </p>
<p>Error, (in asympt) unable to compute series</p><p> </p>
<p>I am interested in finding the asymptofic constant (Big O(1/n^(2m+2)) for the following expansion</p>
<p> </p>
<p><img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=acd54f9d5aec18c25d2b64bb3b8072e6.gif" alt="series((1+1/n)^((1/2)/(sum(1/((2*k+1)*(2*n+1)^(2*k+1)), k = 0 .. m))), n = infinity, 10)"></p>
<p> </p>
<p>Upon using the preceding command in the maple i get</p>
<p> </p>
<p>Error, (in asympt) unable to compute series</p>102406Thu, 10 Mar 2011 20:04:36 Zbarefoot1980barefoot1980