MaplePrimes

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    <title>MaplePrimes - comments on Post, MRB constant L</title>
    <link>http://www.mapleprimes.com/posts/89682-MRB-Constant-L</link>
    <language>en-us</language>
    <copyright>2026 Maplesoft, A Division of Waterloo Maple Inc.</copyright>
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    <lastBuildDate>Thu, 11 Jun 2026 05:10:23 GMT</lastBuildDate>
    <pubDate>Thu, 11 Jun 2026 05:10:23 GMT</pubDate>
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    <itunes:summary />
    <description>The latest comments added to the Post, MRB constant L</description>
    <image>
      <url>http://www.mapleprimes.com/images/mapleprimeswhite.jpg</url>
      <title>MaplePrimes - comments on Post, MRB constant L</title>
      <link>http://www.mapleprimes.com/posts/89682-MRB-Constant-L</link>
    </image>
    <item>
      <title>Infinite values of f are real numbers</title>
      <link>http://www.mapleprimes.com/posts/89682-MRB-Constant-L?ref=Feed:MaplePrimes:MRB constant L:Comments#comment89814</link>
      <itunes:summary>&lt;p&gt;For &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=217f55541dd2628dad490469c6379d85.gif" alt="f(x)=(-1)^x*x^(1/x)"&gt;,&lt;/p&gt;
&lt;p&gt;according to the above graphs, and &lt;a href="http://www.wolframalpha.com/input/?i=+Plot[%28-1%29^x+x^x^%28-1%29%2C+{x%2C+-1%2C+0}]"&gt;this graph from Wolfram|Alpha&lt;/a&gt; there are an infinite number of values, x, from (-1,0) were the values of f are real numbers, having no imaginary part.&lt;/p&gt;
&lt;!--break--&gt;</itunes:summary>
      <description>The latest comments added to the Post, MRB constant L</description>
      <guid>89814</guid>
      <pubDate>Thu, 17 Jun 2010 21:46:12 Z</pubDate>
      <itunes:author>Marvin Ray Burns</itunes:author>
      <author>Marvin Ray Burns</author>
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      <title>the silver ratio</title>
      <link>http://www.mapleprimes.com/posts/89682-MRB-Constant-L?ref=Feed:MaplePrimes:MRB constant L:Comments#comment89864</link>
      <itunes:summary>&lt;p&gt;&lt;span&gt;Again let &lt;img class="math" src="../MapleImage.ashx?f=217f55541dd2628dad490469c6379d85.gif" alt="f(x)=(-1)^x*x^(1/x)"&gt;,&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;For Im(f(x)), the first zero to the right of x=-1 seems to be &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=c08b2e2b63a03eb4c08a73931f8354fa.gif" alt="phi"&gt;-1 where &lt;img class="math" src="../MapleImage.ashx?f=c08b2e2b63a03eb4c08a73931f8354fa.gif" alt="phi"&gt; is the &lt;a href="http://mathworld.wolfram.com/GoldenRatioConjugate.html"&gt;silver ratio&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;Indication that &lt;img class="math" src="../MapleImage.ashx?f=c08b2e2b63a03eb4c08a73931f8354fa.gif" alt="phi"&gt;-1 is most likely a zero:&lt;/p&gt;
&lt;p&gt;&amp;nbsp;evalf(eval(Im(f(x)), x = (sqrt(5)-1)*(1/2)-1))=-1.560570058*10^(-7)&lt;/p&gt;
&lt;p&gt;and&lt;/p&gt;
&lt;p&gt;Digits := 30; evalf(eval(Im(f(x)), x = (sqrt(5)-1)*(1/2)-1))=-1.15017467791296590399352342471*10^(-27)&lt;/p&gt;
&lt;!--break--&gt;
&lt;p&gt;&lt;a href="http://marvinrayburns.com"&gt;&lt;br&gt;&lt;/a&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, MRB constant L</description>
      <guid>89864</guid>
      <pubDate>Fri, 18 Jun 2010 04:23:17 Z</pubDate>
      <itunes:author>Marvin Ray Burns</itunes:author>
      <author>Marvin Ray Burns</author>
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    <item>
      <title>sqrt(3)-2</title>
      <link>http://www.mapleprimes.com/posts/89682-MRB-Constant-L?ref=Feed:MaplePrimes:MRB constant L:Comments#comment89890</link>
      <itunes:summary>&lt;p&gt;&lt;span&gt;
&lt;p&gt;&lt;span&gt;Again  let &lt;img class="math" src="../MapleImage.ashx?f=217f55541dd2628dad490469c6379d85.gif" alt="f(x)=(-1)^x*x^(1/x)"&gt;,&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;For Im(f(x)), the second zero to the right of x=-1 seems to be&amp;nbsp;&lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=b9562c620a8f621166d2d692db7b3d05.gif" alt="sqrt(3)-2"&gt;.&lt;/p&gt;
&lt;p&gt;indication that &lt;span&gt;&lt;img class="math" src="../MapleImage.ashx?f=b9562c620a8f621166d2d692db7b3d05.gif" alt="sqrt(3)-2"&gt;&lt;/span&gt; is a most likely a zero:&lt;/p&gt;
&lt;p&gt;Digits := 10; evalf(eval(Im(f(x)), x = sqrt(3)-2))=-0.000002580544291&lt;br&gt;&amp;nbsp;Digits := 30; evalf(eval(Im(f(x)), x = sqrt(3)-2))= -2.12846188527463303702449027759 10^(-26)&lt;br&gt;...&lt;/p&gt;
&lt;/span&gt;&lt;/p&gt;
&lt;!--break--&gt;
&lt;p&gt;&lt;a href="http://marvinrayburns.com"&gt;&lt;br&gt;&lt;/a&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, MRB constant L</description>
      <guid>89890</guid>
      <pubDate>Fri, 18 Jun 2010 19:14:32 Z</pubDate>
      <itunes:author>Marvin Ray Burns</itunes:author>
      <author>Marvin Ray Burns</author>
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    <item>
      <title>Many more roots</title>
      <link>http://www.mapleprimes.com/posts/89682-MRB-Constant-L?ref=Feed:MaplePrimes:MRB constant L:Comments#comment89897</link>
      <itunes:summary>&lt;p&gt;&lt;span&gt;
&lt;p&gt;&lt;span&gt;
&lt;p&gt;&lt;span&gt;Again  let &lt;img class="math" src="../MapleImage.ashx?f=217f55541dd2628dad490469c6379d85.gif" alt="f(x)=(-1)^x*x^(1/x)"&gt;,&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;For Im(f(x)), the third zero to the right of x=-1 seems to be&lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=8d4db2e99379433aaaa8b0869c016d4b.gif" alt="1/2*(sqrt(21)-5)"&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;.&lt;/p&gt;
&lt;p&gt;Indication that &lt;span&gt;&lt;span&gt;&lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=8d4db2e99379433aaaa8b0869c016d4b.gif" alt="1/2*(sqrt(21)-5)"&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;is a most likely a zero:&lt;/p&gt;
&lt;p&gt;Digits := 10; evalf(eval(Im(f(x)), x = 1/2*(sqrt(21)-5)))=0.00006274148290&lt;br&gt;Digits := 30; evalf(eval(Im(f(x)), x = 1/2*(sqrt(21)-5)))=-5.15516567057872417416934365318*10^(-25)&lt;br&gt;...&lt;/p&gt;
&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;The fouth zero &lt;span&gt;&lt;span&gt;seems to be&lt;/span&gt;&lt;/span&gt; &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=84282cc97ca28e9bfee025bdd1ee928b.gif" alt="srt(2)-1"&gt;.&lt;/p&gt;
&lt;p&gt;The fifth zero &lt;span&gt;&lt;span&gt;seems to be&lt;/span&gt;&lt;/span&gt; &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=cf00555a224d02f9b0a21b6d0b0ccc0b.gif" alt="1/2*(sqrt(357)-19)"&gt;.&lt;/p&gt;
&lt;p&gt;The sixth zero &lt;span&gt;&lt;span&gt;seems to be&lt;/span&gt;&lt;/span&gt; &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=b9b132972705e22a865fe023200b6ee6.gif" alt="12*sqrt(2)-17"&gt;.&lt;/p&gt;
&lt;p&gt;The seventh zero&lt;span&gt;&lt;span&gt; seems to be&lt;/span&gt;&lt;/span&gt; &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=e54c60468cdcfcc82993e7cbe74f9dfe.gif" alt="6*sqrt(10)-19"&gt;.&lt;/p&gt;
&lt;p&gt;To check &lt;img class="math" src="../MapleImage.ashx?f=e54c60468cdcfcc82993e7cbe74f9dfe.gif" alt="6*sqrt(10)-19"&gt;:&lt;/p&gt;
&lt;p&gt;Digits := 80; evalf(eval(Im(f(x)), x = 6*sqrt(10)-19))=1.4149889409...*10^(-15)&lt;br&gt;Digits := 160; evalf(eval(Im(f(x)), x = 6*sqrt(10)-19))=-2.034233693511...10^(-96)&lt;/p&gt;
&lt;p&gt;...&lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;The next zero seems to be &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=28fa1fd2293e333d709ce277233cd3e7.gif" alt="4*sqrt(33)-23"&gt;&lt;/p&gt;
&lt;p&gt;Next &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=ba5d7e0176ac14767d14fda05ff75e83.gif" alt="5*sqrt(23)-24"&gt;&lt;/p&gt;
&lt;p&gt;Next &lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=3deeb3fbee3a1d5bc79f01aa8219f4fb.gif" alt="4*sqrt(39)-25"&gt;&lt;/p&gt;
&lt;p&gt;To check &lt;img class="math" src="../MapleImage.ashx?f=3deeb3fbee3a1d5bc79f01aa8219f4fb.gif" alt="4*sqrt(39)-25"&gt;&lt;/p&gt;
&lt;p&gt;Digits := 160; evalf(eval(Im(f(x)), x = 4*sqrt(39)-25))=-2.5063583022939...*10^(-70)&lt;br&gt;Digits := 200; evalf(eval(Im(f(x)), x = 4*sqrt(39)-25))=-4.02129897436313282..*10^(-111)&lt;/p&gt;
&lt;p&gt;...&lt;/p&gt;
&lt;!--break--&gt;
&lt;p&gt;&lt;a href="http://marvinrayburns.com"&gt;marvinrayburns.com&lt;br&gt;&lt;/a&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, MRB constant L</description>
      <guid>89897</guid>
      <pubDate>Fri, 18 Jun 2010 22:41:31 Z</pubDate>
      <itunes:author>Marvin Ray Burns</itunes:author>
      <author>Marvin Ray Burns</author>
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      <title>All of the zeros</title>
      <link>http://www.mapleprimes.com/posts/89682-MRB-Constant-L?ref=Feed:MaplePrimes:MRB constant L:Comments#comment89916</link>
      <itunes:summary>&lt;!--break--&gt;
&lt;p&gt;&lt;a href="http://marvinrayburns.com"&gt;&lt;/a&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;Again   let &lt;img class="math" src="../MapleImage.ashx?f=217f55541dd2628dad490469c6379d85.gif" alt="f(x)=(-1)^x*x^(1/x)"&gt;.&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;All of the zeros&lt;/strong&gt;&lt;strong&gt; &amp;isin; &amp;lt;&lt;/strong&gt;ℝ&lt;strong&gt;,&lt;/strong&gt;&lt;strong&gt;[-1,0)&amp;gt; of Im(f(x)) are the roots &lt;/strong&gt;&lt;strong&gt;&amp;isin; &amp;lt;&lt;/strong&gt;ℝ&lt;strong&gt;,&lt;/strong&gt;&lt;strong&gt;[-1,0)&amp;gt;&lt;/strong&gt;&lt;strong&gt; of x^2+W*x=-1, were W &amp;isin;&amp;lt;&lt;/strong&gt;ℕ&lt;strong&gt; ,[2,&amp;infin;)&amp;gt;.&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;I will try to put it this way:&lt;/p&gt;
&lt;p&gt;{ zeros &amp;isin; &amp;lt;ℝ,[-1,0)&amp;gt;  of Im(f(x)) } = { roots &amp;isin; &amp;lt;ℝ,[-1,0)&amp;gt; of x^2+W*x=-1 }, were W &amp;isin;&amp;lt;ℕ ,[2,&amp;infin;)&amp;gt;.&lt;strong&gt;&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;&lt;strong&gt;The two sets are equal!&lt;br&gt;&lt;/strong&gt;&lt;/p&gt;
&lt;p&gt;&lt;span&gt;ℕ stands for natural numbers.&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;span&gt;s &lt;/span&gt;&amp;isin; &amp;lt;ℝ,[-1,0)&amp;gt; &lt;span&gt;means s is in the set of all real numbers &amp;gt;=-1 and &amp;lt;0 .&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&lt;span&gt;W  &amp;isin;&amp;lt;ℕ ,(1,&amp;infin;)&amp;gt; means W is in the set of all natural numbers greater than 1.&lt;strong&gt;&lt;br&gt;&lt;/strong&gt;&lt;/span&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, MRB constant L</description>
      <guid>89916</guid>
      <pubDate>Sun, 20 Jun 2010 00:42:54 Z</pubDate>
      <itunes:author>Marvin Ray Burns</itunes:author>
      <author>Marvin Ray Burns</author>
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      <title>The sum of the zeros</title>
      <link>http://www.mapleprimes.com/posts/89682-MRB-Constant-L?ref=Feed:MaplePrimes:MRB constant L:Comments#comment89930</link>
      <itunes:summary>&lt;p&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;span&gt;Again    let &lt;img class="math" src="../MapleImage.ashx?f=217f55541dd2628dad490469c6379d85.gif" alt="f(x)=(-1)^x*x^(1/x)"&gt;.&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;I had hoped to find some combination of known constants to equal the &lt;span&gt; sum of the zeros  &amp;isin; &amp;lt;ℝ,[-1,0)&amp;gt;  of Im(f(x), but,&lt;/span&gt;&lt;span&gt; the sum of the zeros  &amp;isin; &amp;lt;ℝ,[-1,0)&amp;gt;  of Im(f(x)) is (very slowly) divergent, as in the following:&lt;br&gt;&lt;/span&gt;&lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;&lt;span&gt; &lt;br&gt; &lt;br&gt; &lt;/span&gt;&lt;/p&gt;
&lt;form&gt; &lt;/form&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(1)&lt;/td&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(2)&lt;/td&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(3)&lt;/td&gt;
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&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;x=1/2 (-W + sqrt[-4 + W^2])&lt;/p&gt;
&lt;p&gt;limit(1/2 *(-W + sqrt(-4 + W^2)),W= infinity)=&lt;img class="math" src="http://www.mapleprimes.com/MapleImage.ashx?f=6777da1af1e0e7527531e3f43ef27cfb.gif" alt="limit(1/2*(-W+sqrt(-4+W^2)),W= infinity)"&gt;&lt;/p&gt;
&lt;p&gt;and&amp;nbsp;sum(1/2 *(-W + sqrt(-4 + W^2)),W=2 ..infinity) = -infinity&lt;/p&gt;
&lt;p&gt;&amp;nbsp;&lt;/p&gt;
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&lt;p&gt;&lt;span&gt;&amp;nbsp;&lt;/span&gt;&lt;/p&gt;
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&lt;p&gt;&lt;br&gt; &lt;br&gt;&lt;/p&gt;
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      <description>The latest comments added to the Post, MRB constant L</description>
      <guid>89930</guid>
      <pubDate>Sun, 20 Jun 2010 17:14:39 Z</pubDate>
      <itunes:author>Marvin Ray Burns</itunes:author>
      <author>Marvin Ray Burns</author>
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