Markus–Yamabe conjecture: Difference between revisions
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{{for|the generalization of hypergeometric series |MacRobert E function}} | |||
In [[mathematics]], '''E-functions''' are a type of [[power series]] that satisfy particular arithmetic conditions on the coefficients. They are of interest in [[transcendence theory]], and are more special than [[G-function (power series)|G-function]]s. | |||
==Definition== | |||
A function ''f''(''x'') is called of '''type ''E''''', or an '''''E''-function''',<ref>Carl Ludwig Siegel, ''Transcendental Numbers'', p.33, Princeton University Press, 1949.</ref> if the power series | |||
:<math>f(x)=\sum_{n=0}^{\infty} c_{n}\frac{x^{n}}{n!}</math> | |||
satisfies the following three conditions: | |||
* All the coefficients ''c<sub>n</sub>'' belong to the same [[algebraic number field]], ''K'', which has [[Degree of a field extension|finite degree]] over the rational numbers; | |||
* For all ε > 0, | |||
:<math>\overline{\left|c_{n}\right|}=O\left(n^{n\varepsilon}\right)</math>, | |||
where the left hand side represents the maximum of the absolute values of all the [[Conjugate element (field theory)|algebraic conjugates]] of ''c<sub>n</sub>''; | |||
* For all ε > 0 there is a sequence of natural numbers ''q''<sub>0</sub>, ''q''<sub>1</sub>, ''q''<sub>2</sub>,… such that ''q<sub>n</sub>c<sub>k''</sub> is an [[algebraic integer]] in ''K'' for ''k''=0, 1, 2,…, ''n'', and ''n'' = 0, 1, 2,… and for which | |||
:<math>q_{n}=O\left(n^{n\varepsilon}\right)</math>. | |||
The second condition implies that ''f'' is an [[entire function]] of ''x''. | |||
==Uses== | |||
''E''-functions were first studied by [[Carl Ludwig Siegel|Siegel]] in 1929.<ref>C.L. Siegel, ''Über einige Anwendungen diophantischer Approximationen'', Abh. Preuss. Akad. Wiss. '''1''', 1929.</ref> He found a method to show that the values taken by certain ''E''-functions were [[algebraically independent]].This was a result which established the algebraic independence of classes of numbers rather than just linear independence.<ref>Alan Baker, ''Transcendental Number Theory'', pp.109-112, Cambridge University Press, 1975.</ref> Since then these functions have proved somewhat useful in [[number theory]] and in particular they have application in [[Transcendental numbers|transcendence]] proofs and [[differential equations]].<ref>Serge Lang, ''Introduction to Transcendental Numbers'', pp.76-77, Addison-Wesley Publishing Company, 1966.</ref> | |||
==The Siegel–Shidlovsky theorem== | |||
Perhaps the main result connected to ''E''-functions is the Siegel–Shidlovsky theorem (also known as the Shidlovsky and Shidlovskii theorem), named after [[Carl Ludwig Siegel]] and Andrei Borisovich Shidlovskii. | |||
Suppose that we are given ''n'' ''E''-functions, ''E''<sub>1</sub>(''x''),…,''E''<sub>''n''</sub>(''x''), that satisfy a system of homogeneous linear differential equations | |||
:<math>y^\prime_i=\sum_{j=1}^n f_{ij}(x)y_j\quad(1\leq i\leq n)</math> | |||
where the ''f<sub>ij</sub>'' are rational functions of ''x'', and the coefficients of each ''E'' and ''f'' are elements of an algebraic number field ''K''. Then the theorem states that if ''E''<sub>1</sub>(''x''),…,''E''<sub>''n''</sub>(''x'') are algebraically independent over ''K''(''x''), then for any non-zero algebraic number α that is not a pole of any of the ''f<sub>ij</sub>'' the numbers ''E''<sub>1</sub>(α),…,''E''<sub>''n''</sub>(α) are algebraically independent. | |||
==Examples== | |||
# Any polynomial with algebraic coefficients is a simple example of an ''E''-function. | |||
# The [[exponential function]] is an ''E''-function, in its case ''c<sub>n</sub>''=1 for all of the ''n''. | |||
# If λ is an algebraic number then the [[Bessel function]] ''J''<sub>λ</sub> is an ''E''-function. | |||
# The sum or product of two ''E''-functions is an ''E''-function. In particular ''E''-functions form a [[Ring (mathematics)|ring]]. | |||
# If ''a'' is an algebraic number and ''f''(''x'') is an ''E''-function then ''f''(''ax'') will be an ''E''-function. | |||
# If ''f''(''x'') is an ''E''-function then the derivative and integral of ''f'' are also ''E''-functions. | |||
==References== | |||
<references/> | |||
* {{mathworld|title=E-Function|urlname=E-Function}} | |||
[[Category:Number theory]] | |||
Latest revision as of 16:55, 20 April 2013
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendence theory, and are more special than G-functions.
Definition
A function f(x) is called of type E, or an E-function,[1] if the power series
satisfies the following three conditions:
- All the coefficients cn belong to the same algebraic number field, K, which has finite degree over the rational numbers;
- For all ε > 0,
where the left hand side represents the maximum of the absolute values of all the algebraic conjugates of cn;
- For all ε > 0 there is a sequence of natural numbers q0, q1, q2,… such that qnck is an algebraic integer in K for k=0, 1, 2,…, n, and n = 0, 1, 2,… and for which
The second condition implies that f is an entire function of x.
Uses
E-functions were first studied by Siegel in 1929.[2] He found a method to show that the values taken by certain E-functions were algebraically independent.This was a result which established the algebraic independence of classes of numbers rather than just linear independence.[3] Since then these functions have proved somewhat useful in number theory and in particular they have application in transcendence proofs and differential equations.[4]
The Siegel–Shidlovsky theorem
Perhaps the main result connected to E-functions is the Siegel–Shidlovsky theorem (also known as the Shidlovsky and Shidlovskii theorem), named after Carl Ludwig Siegel and Andrei Borisovich Shidlovskii.
Suppose that we are given n E-functions, E1(x),…,En(x), that satisfy a system of homogeneous linear differential equations
where the fij are rational functions of x, and the coefficients of each E and f are elements of an algebraic number field K. Then the theorem states that if E1(x),…,En(x) are algebraically independent over K(x), then for any non-zero algebraic number α that is not a pole of any of the fij the numbers E1(α),…,En(α) are algebraically independent.
Examples
- Any polynomial with algebraic coefficients is a simple example of an E-function.
- The exponential function is an E-function, in its case cn=1 for all of the n.
- If λ is an algebraic number then the Bessel function Jλ is an E-function.
- The sum or product of two E-functions is an E-function. In particular E-functions form a ring.
- If a is an algebraic number and f(x) is an E-function then f(ax) will be an E-function.
- If f(x) is an E-function then the derivative and integral of f are also E-functions.
References
- ↑ Carl Ludwig Siegel, Transcendental Numbers, p.33, Princeton University Press, 1949.
- ↑ C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. 1, 1929.
- ↑ Alan Baker, Transcendental Number Theory, pp.109-112, Cambridge University Press, 1975.
- ↑ Serge Lang, Introduction to Transcendental Numbers, pp.76-77, Addison-Wesley Publishing Company, 1966.
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