![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/17/MRB_messy.gif/195px-MRB_messy.gif)
This mathematical constant is sometimes called the MRB constant[1][2][3][4][5] or MRB..[6] MRB stands for Marvin Ray Burns. Being a sum of irrational numbers its irrationally remains an open problem.[7]
The numerical value of the constant, truncated to 6 decimal places, is
Definition
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/MRB-Gif.gif/195px-MRB-Gif.gif)
The constant is related to the following divergent series:
Its partial sums
are bounded so that their limit points form an interval [−0.812140…,0.187859…] of length 1..[8] The upper limit point 0.187859… is what is sometimes called the MRB constant. [9] The constant can be explicitly defined by the following infinite sums: [10] [11]
There is no known closed-form expression of this constant.[12]
History
editMarvin Ray Burns published his discovery of the constant in 1999.[13] The discovery is a result of a "math binge" that started in the spring of 1994.[14] Before verifying with colleague Simon Plouffe that such a constant had not already been discovered or at least not widely published, Burns called the constant "rc" for root constant.[15] At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999.[16] Since then it has been added to tables of constants in a few countries, including Turkey,[17] Iran,[18] Germany.[19] and the United States.[20]
References
edit- ^ Weisstein, Eric W. ""MRB Constant". MathWorld. MathWorld--A Wolfram Web Resource. Retrieved 12 January 2015.
- ^ MATHAR, RICHARD J (2009). "NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(iπx) x^*1/x) BETWEEN 1 AND INFINITY" (PDF). Cornell University. arXiv:0912.3844v3. Retrieved 12 January 2015.
- ^ Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). marvinrayburns.com. PSI Press. Retrieved 13 January 2015.
- ^ (sequence A160755 in the OEIS) and (sequence A173273 in the OEIS)
- ^ Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Mauro Fiorentini. Retrieved 14 January 2015.
- ^ Finch, Steven R. "Irrationality of the MRB constsnt". marvinrayburns.com. Marvin Burns. Retrieved 13 January 2015.
- ^ Finch, Steven R. "Irrationality of the MRB constsnt". marvinrayburns.com. Marvin Burns. Retrieved 13 January 2015.
- ^ Weisstein, Eric W. ""MRB Constant". MathWorld. MathWorld--A Wolfram Web Resource. Retrieved 12 January 2015.
- ^ Weisstein, Eric W. ""MRB Constant". MathWorld. MathWorld--A Wolfram Web Resource. Retrieved 12 January 2015.
- ^ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.
- ^ Weisstein, Eric W. "MRB Constant". MathWorld.
- ^ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.
- ^ Burns, Marvin. "mrburns". plouffe.fr. SImeon Plouffe. Retrieved 12 January 2015.
- ^ Burns, Marvin R. (2002-04-12). "Captivity's Captor: Now is the Time for the Chorus of Conversion". Indiana University. Retrieved 2009-05-05.
- ^ Burns, Marvin R. (1999-01-23). "RC". math2.org. Retrieved 2009-05-05.
{{cite web}}
: External link in
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- ^ Plouffe, Simon (1999-11-20). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 2009-05-05.
{{cite web}}
: External link in
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- ^ "Matematıksel Sabıtler" (in Turkish). Türk Biyofizik Derneği. Retrieved 2015-01-13.
{{cite web}}
: External link in
(help)CS1 maint: unrecognized language (link)|publisher=
- ^ "Mathematical Constants". Iran Civil Center. Archived from the original on 2008-11-21. Retrieved 2015-01-13.
{{cite web}}
: External link in
(help)|publisher=
- ^ "Etymologie CA Kanada Zahlen" (in German). etymologie.info. Retrieved 2015-01-13.
{{cite web}}
: External link in
(help)CS1 maint: unrecognized language (link)|publisher=
- ^ Weisstein, Eric W. "MiscellaneousConstants". mathworld.wolfram.com. Wolfram Research. Retrieved 14 January 2015.
External links
editCategory:Mathematical constants Category:Number theory
![](http://upload.wikimedia.org/wikipedia/commons/thumb/1/17/MRB_messy.gif/195px-MRB_messy.gif)
The MRB constant, named after Marvin Ray Burns, is a mathematical constant for which no closed-form expression is known. It is not known whether the MRB constant is algebraic, transcendental, or even irrational.
The numerical value of MRB constant, truncated to 6 decimal places, is
Definition
edit![](http://upload.wikimedia.org/wikipedia/commons/thumb/8/8c/MRB-Gif.gif/195px-MRB-Gif.gif)
The MRB constant is related to the following divergent series:
Its partial sums
are bounded so that their limit points form an interval [−0.812140…,0.187859…] of length 1. The upper limit point 0.187859… is what is known as the MRB constant.[1][2][3][4][5][6][7]
The MRB constant can be explicitly defined by the following infinite sums:[8]
There is no known closed-form expression of the MRB constant.[9]
History
editHistory Marvin Ray Burns published his discovery of the constant in 1999.[10] The discovery is a result of a "math binge" that started in the spring of 1994.[11] Before verifying with colleague Simon Plouffe that such a constant had not already been discovered or at least not widely published, Burns called the constant "rc" for root constant.[12] At Plouffe's suggestion, the constant was renamed Marvin Ray Burns's Constant, and then shortened to "MRB constant" in 1999.[13] Since then it has been added to tables and lists of constants in a few countries, including Turkey,[14] Iran,[15] Germany.[16] , the United States.[17] and Italy[18]
References
edit- ^ Weisstein, Eric W. ""MRB Constant". MathWorld. MathWorld--A Wolfram Web Resource. Retrieved 12 January 2015.
- ^ MATHAR, RICHARD J (2009). "NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(iπx) x^*1/x) BETWEEN 1 AND INFINITY" (PDF). Cornell University. arXiv:0912.3844v3. Retrieved 12 January 2015.
- ^ Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). PSI Press. Archived from the original (PDF) on 2013-04-30. Retrieved 16 January 2015.
- ^ (sequence A037077 in the OEIS)
- ^ (sequence A160755 in the OEIS)
- ^ (sequence A173273 in the OEIS)
- ^ Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Mauro Fiorentini. Retrieved 14 January 2015.
- ^ Weisstein, Eric W. "MRB Constant". MathWorld.
- ^ Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.
- ^ Burns, Marvin. "mrburns". plouffe.fr. SImeon Plouffe. Retrieved 12 January 2015.
- ^ Burns, Marvin R. (2002-04-12). "Captivity's Captor: Now is the Time for the Chorus of Conversion". Indiana University. Retrieved 2009-05-05.
- ^ Burns, Marvin R. (1999-01-23). "RC". math2.org. Retrieved 2009-05-05.
{{cite web}}
: External link in
(help)|publisher=
- ^ Plouffe, Simon (1999-11-20). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 2009-05-05.
{{cite web}}
: External link in
(help)|publisher=
- ^ "Matematıksel Sabıtler" (in Turkish). Türk Biyofizik Derneği. Retrieved 2015-01-13.
{{cite web}}
: External link in
(help)CS1 maint: unrecognized language (link)|publisher=
- ^ "Mathematical Constants". Iran Civil Center. Archived from the original on 2008-11-21. Retrieved 2015-01-13.
{{cite web}}
: External link in
(help)|publisher=
- ^ "Etymologie CA Kanada Zahlen" (in German). etymologie.info. Retrieved 2015-01-13.
{{cite web}}
: External link in
(help)CS1 maint: unrecognized language (link)|publisher=
- ^ Weisstein, Eric W. "MiscellaneousConstants". mathworld.wolfram.com. Wolfram Research. Retrieved 14 January 2015.
- ^ Sýkora, Stanislav. "Mathematical Constants". ebyte.it. ebyte.it. Retrieved 1 February 2015.
External links
edit
Iterated exponentials are an example of an iterated function system based on . Such systems have induced some interesting mathematical constants and interesting fractal properties based on its generalization to the complex plane.Cite error: A <ref>
tag is missing the closing </ref>
(see the help page).
Inverse
editIn fact, does have an inverse
which is well-defined for
This has induced interest in the function , which has similar limiting properties to . [1]
Convergence
editBy an old result of Euler, repeated exponentiation convergence for real values inbetween and .[2]
Calculation of Iterated Exponential
editIn certain situations, one may calculate the iterated exponential, and certain constants remain of mathematical interest.
Connection to Lambert's Function
editIf one defines
for such where such a process converges,
Then actually has a closed form expression in terms of a function known as Lambert's function which is defined implicitly via the following equation:
Namely, that
This can be seen by inputting this definition of into the other equation that satisfies, . [3]
Iteration on the Complex Plane
editThe function may also be extended to the complex plane, where such a map tends to display interesting fractal properties.[4]
Of particular interest is evaluation of the constant
Which does indeed converge [5] and has been evaluated as
<ref>Galidakis, I. N. (2004). On an application of Lambert's W function to infinite exponentials. Complex Variables, Theory and Application: An International Journal, 49(11), 759-780.</math>
Category:Mathematical constants
Category:Number theory
- ^ De Villiers, J. M., & Robinson, P. N. (1986). The interval of convergence and limiting functions of a hyperpower sequence. American Mathematical Monthly, 13-23.
- ^ L. Euler, De formulis exponentialibus replicatis, Leonhardi Euleri Opera Omnia, Ser. 1, Opera Mathematica 15 (1927) 268-297
- ^ Corless, R. M., Gonnet, G. H., Hare, D. E., Jeffrey, D. J., & Knuth, D. E. (1996). On the Lambert W function. Advances in Computational mathematics, 5(1), 329-359.
- ^ Baker, I. N., & Rippon, P. J. (1985). A note on complex iteration. American Mathematical Monthly, 501-504.
- ^ Macintyre, A. J. (1966). Convergence of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle 𝑖^{𝑖𝑖 \cdots}} . Proceedings of the American Mathematical Society, 17(1), 67.