Hypertranscendental function: Difference between revisions
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{{Short description|Mathematics analytic function}} |
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{{Use American English|date = January 2019}} |
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| ⚫ | A '''hypertranscendental function''' or '''transcendentally transcendental function''' is a [[transcendental function|transcendental]] [[analytic function]] which is not the solution of an [[algebraic differential equation]] with coefficients in <math>\mathbb{Z}</math> (the [[integer]]s) and with algebraic [[initial condition]]s. |
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==History== |
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The term was introduced by [[D. D. Morduhai-Boltovskoi]] in 1949.<ref>{{harvtxt|Morduhaĭ-Boltovskoĭ|1949}}</ref> |
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The term 'transcendentally transcendental' was introduced by [[E. H. Moore]] in 1896; the term 'hypertranscendental' was introduced by [[D. D. Morduhai-Boltovskoi]] in 1914.<ref>D. D. Mordykhai-Boltovskoi, "On hypertranscendence of the function ξ(x, s)", ''Izv. Politekh. Inst. Warsaw''<!-- what is the full name of this journal? presumably something like Izwiestija Politechniki Instituti Warszawskiej, but (a) I don't know Polish (b) I haven't been able to find a journal of that name or similar--> '''2''':1-16 (1914), cited in Anatoly A. Karatsuba, S. M. Voronin, ''The Riemann Zeta-Function'', 1992, {{ISBN|3-11-013170-6}}, [https://books.google.com/books?id=fNontpCu9kQC&pg=PA390 p. 390]</ref><ref>{{harvtxt|Morduhaĭ-Boltovskoĭ|1949}}</ref> |
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==Definition== |
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One standard definition (there are slight variants) defines solutions of [[differential equation]]s of the form |
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:<math>F\left(x, y, y', \cdots, y^{(n)} \right) = 0</math>, |
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where <math>F</math> is a polynomial with constant coefficients, as ''algebraically transcendental'' or ''differentially algebraic''. Transcendental functions which are not ''algebraically transcendental'' are ''transcendentally transcendental''. [[Hölder's theorem]] shows that the [[gamma function]] is in this category.<ref>[[Eliakim H. Moore]], "Concerning Transcendentally Transcendental Functions", ''Mathematische Annalen'' '''48''':1-2:49-74 (1896) {{doi|10.1007/BF01446334}}</ref><ref>[[Robert Daniel Carmichael|R. D. Carmichael]], "On Transcendentally Transcendental Functions", ''Transactions of the American Mathematical Society'' '''14''':3:311-319 (July 1913) [https://www.ams.org/journals/tran/1913-014-03/S0002-9947-1913-1500949-2/S0002-9947-1913-1500949-2.pdf full text] {{JSTOR|1988599}} {{doi|10.1090/S0002-9947-1913-1500949-2}}</ref><ref>Lee A. Rubel, "A Survey of Transcendentally Transcendental Functions", ''The American Mathematical Monthly'' '''96''':777-788 (November 1989) {{JSTOR|2324840}}</ref> |
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==Examples== |
==Examples== |
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===Hypertranscendental functions=== |
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* The zeta functions of [[algebraic number field]]s, in particular, the [[Riemann zeta function]] |
* The zeta functions of [[algebraic number field]]s, in particular, the [[Riemann zeta function]] |
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* The [[ |
* The [[gamma function]] (''cf.'' [[Hölder's theorem]]) |
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* The [[exponential function]], [[logarithm]], and the [[trigonometric function|trigonometric]] and [[hyperbolic function|hyperbolic]] functions. |
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* The [[generalized hypergeometric function]]s, including special cases such as [[Bessel function]]s (except some special cases which are algebraic). |
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===Non-transcendental (algebraic) functions=== |
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* All [[algebraic function]]s, in particular [[polynomial]]s. |
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* Any [[polynomial]] with algebraic coefficients |
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* The [[exponential function]] and the [[logarithm]] |
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* The sine, cosine and tangent [[trigonometric function]]s |
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==See also== |
==See also== |
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==References== |
==References== |
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* Loxton,J.H., Poorten,A.J. van der, "[http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN356261603_0016 A class of hypertranscendental functions]", |
* Loxton, J.H., Poorten, A.J. van der, "[http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN356261603_0016 A class of hypertranscendental functions]", [[Aequationes Mathematicae]], Periodical volume 16 |
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* [[Kurt Mahler|Mahler,K.]], "Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen", Math. Z. 32 (1930) 545-585. |
* [[Kurt Mahler|Mahler, K.]], "Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen", Math. Z. 32 (1930) 545-585. |
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* {{citation|mr=0028347|last=Morduhaĭ-Boltovskoĭ|first=D.|title=On hypertranscendental functions and hypertranscendental numbers|language=Russian|journal=Doklady |
* {{citation|mr=0028347|last=Morduhaĭ-Boltovskoĭ|first=D.|title=On hypertranscendental functions and hypertranscendental numbers|language=Russian|journal=Doklady Akademii Nauk SSSR |series=New Series|volume=64|year=1949|pages= 21–24}} |
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[[Category:Analytic functions]] |
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[[Category:Mathematical analysis]] |
[[Category:Mathematical analysis]] |
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[[Category:Types of functions]] |
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[[Category:Ordinary differential equations]] |
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Latest revision as of 22:35, 27 June 2024
A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in (the integers) and with algebraic initial conditions.
History
[edit]The term 'transcendentally transcendental' was introduced by E. H. Moore in 1896; the term 'hypertranscendental' was introduced by D. D. Morduhai-Boltovskoi in 1914.[1][2]
Definition
[edit]One standard definition (there are slight variants) defines solutions of differential equations of the form
- ,
where is a polynomial with constant coefficients, as algebraically transcendental or differentially algebraic. Transcendental functions which are not algebraically transcendental are transcendentally transcendental. Hölder's theorem shows that the gamma function is in this category.[3][4][5]
Hypertranscendental functions usually arise as the solutions to functional equations, for example the gamma function.
Examples
[edit]Hypertranscendental functions
[edit]- The zeta functions of algebraic number fields, in particular, the Riemann zeta function
- The gamma function (cf. Hölder's theorem)
Transcendental but not hypertranscendental functions
[edit]- The exponential function, logarithm, and the trigonometric and hyperbolic functions.
- The generalized hypergeometric functions, including special cases such as Bessel functions (except some special cases which are algebraic).
Non-transcendental (algebraic) functions
[edit]- All algebraic functions, in particular polynomials.
See also
[edit]Notes
[edit]- ^ D. D. Mordykhai-Boltovskoi, "On hypertranscendence of the function ξ(x, s)", Izv. Politekh. Inst. Warsaw 2:1-16 (1914), cited in Anatoly A. Karatsuba, S. M. Voronin, The Riemann Zeta-Function, 1992, ISBN 3-11-013170-6, p. 390
- ^ Morduhaĭ-Boltovskoĭ (1949)
- ^ Eliakim H. Moore, "Concerning Transcendentally Transcendental Functions", Mathematische Annalen 48:1-2:49-74 (1896) doi:10.1007/BF01446334
- ^ R. D. Carmichael, "On Transcendentally Transcendental Functions", Transactions of the American Mathematical Society 14:3:311-319 (July 1913) full text JSTOR 1988599 doi:10.1090/S0002-9947-1913-1500949-2
- ^ Lee A. Rubel, "A Survey of Transcendentally Transcendental Functions", The American Mathematical Monthly 96:777-788 (November 1989) JSTOR 2324840
References
[edit]- Loxton, J.H., Poorten, A.J. van der, "A class of hypertranscendental functions", Aequationes Mathematicae, Periodical volume 16
- Mahler, K., "Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen", Math. Z. 32 (1930) 545-585.
- Morduhaĭ-Boltovskoĭ, D. (1949), "On hypertranscendental functions and hypertranscendental numbers", Doklady Akademii Nauk SSSR, New Series (in Russian), 64: 21–24, MR 0028347