Śleszyński–Pringsheim theorem
The convergence criterion of Pringsheim or main criterion of Pringsheim is a criterion on the convergence behavior of infinite continued fractions. It goes back to the German mathematician Alfred Pringsheim and one of the classic tenets of the continued fraction theory within the analytic number theory. In the English literature, the criterion theorem under the name Śleszyński - Pringsheim 's ( and similar) is guided, the former name of the Polish- Russian mathematician Ivan Śleszyński ( 1854-1931 ) indicates that this criterion also and before Pringsheim had found. There is evidence that Alfred Pringsheim appropriate publication of Ivan Śleszyński possibly knew when he made his publication in 1898. It must be added here but also Notice of Oskar Perron in Volume II of his theory of continued fractions, after which the substance is to find this set already in the textbook of algebraic Analysis of Moritz Abraham Stern (Leipzig 1860).
- 2.1 Conclusion I: The set of Worpitzky
- 2.2 Conclusion II: More convergence criterion of Pringsheim
- 3.1 set of Star pride
- 3.2 set of Seidel- Stern
- 3.3 convergence theorem of Tietze 3.3.1 related to irrationality
- 4.1 Example I
- 4.2 Example II
- 4.3 Example III
- 4.4 Example IV
- 4.5 Counter-example
- 4.6 Application: representation of real numbers by negative - regular continued fractions 4.6.1 Formulation of the representation theorem
- 4.6.2 Additional I: algorithm to determine the partial denominators
- 4.6.3 Addition II: distinction of rational and irrational numbers
- 4.6.4 Examples of negative - regular continued fraction representations
Formulation of the criterion
Part I
For two sequences of numbers and complex numbers with the property that the inequalities
Are met, is the associated continued fraction
Always convergent. This means:
The consequence of the proximity breaks
Is a convergent sequence and through them clearly certain limit with
Is the value of the associated continued fraction.
Part II
In the event that the above condition is satisfied, shall always
Part III
The limiting case occurs if and only if the following three conditions are satisfied:
In this limiting case, the chain has a value of fracture
Conclusions
From the Pringsheim convergence criterion of the several other convergence criteria can be derived. These include the following:
Corollary I: The set of Worpitzky
For a number sequence of complex numbers, which in all sequence elements the inequality
Fulfilled, the continued fraction
Always convergent.
The following applies for the convergents always
And accordingly the value of the continued fraction
The set of Worpitzky was published in 1865 by Julius Worpitzky and is considered the first convergence criterion for continued fractions with elements of the complex plane.
Corollary II: More convergence criterion of Pringsheim
Through specialization can be found with the convergence criterion of Pringsheim another, which Alfred Pringsheim has formulated in his work On the convergence of infinite continued fractions in the meeting reports of the Bavarian Academy of Sciences from 1898 itself and which reads as follows:
For a number sequence of complex numbers, which in all sequence elements the inequality
Met, the regular continued fraction
Always convergent.
This further convergence criterion of Pringsheim, for example, always applicable to the case that all partial denominators have at least the amount 2.
Related criteria: the sets of Star Pride and Seidel- Stern, and the convergence theorem of Tietze
In the case of regular infinite continued fractions, there are several criteria that come into play as a supplement to pringsheimschen convergence criterion repeatedly on the issue of convergence and divergence. These include the rates shown below, which include in addition to the classical results of this continued fraction convergence theory.
Set of Star pride
The set of Star pride formulated a very general condition for the divergence of infinite regular continued fractions and reads as follows:
An arbitrary complex continued fraction
To a number sequence of complex numbers
In each case is divergent, if the corresponding row
Is absolutely convergent. That is to say: For the convergence of the chain breaking, it is always necessary that
Applies.
This criterion goes back to Moritz Abraham Stern and Otto pride.
Set of Seidel- Stern
The set of Seidel- Stern tightened the set of Star pride for the case of regular infinite continued fractions with denominators consistently positive part head by presenting the latter condition even as a necessary and sufficient condition. It runs thus:
For a number sequence of positive real numbers converges to the continued fraction
If and only if the associated series
Diverges.
This criterion goes back to Philipp Ludwig von Seidel and Moritz Abraham Stern. It comes into play when said in Part I of pringsheimschen criterion inequality is not consistently achievable, but can be replaced in conjunction with the assumed positivity of the partial denominators through the series divergence condition.
Convergence theorem of Tietze
The convergence theorem of Tietze also discussed the convergence of an infinite continued fractions. He goes back to the German mathematician Heinrich Tietze and states the following:
There are two numerical sequences of real numbers and, which satisfy the following three conditions for all indices:
Then the associated continued fraction
Always convergent. The consequence of the proximity breaks
Converges in this case to the limit
And this applies
Or
Moreover, the denominators of the convergents always satisfy the inequality
And it's
Associated with irrationality
Based on the convergence theorem of Tietze can be achieved Irrationalitätsaussagen. As already proved Heinrich Tietze itself converges every infinite continued fraction of the form ( *) always - with one exception! - Against an irrational number, if you tightened the conditions as follows:
The exception to this is present when from an index in addition to all indices, the following exception condition (A ) is satisfied:
In this exceptional case, the limit value is a rational number.
Examples and Application
Example I
The convergence criterion of the following Pringsheim convergent infinite continued fraction:
As (IIIb) is not satisfied, part III is not applicable. Rather, it is
As is clear from the continued fractions found by Leonhard Euler and Ernesto Cesàro the Eulerian number. Therefore, because of the transcendence of Euler's number, the number is also a transcendental number.
Example II
The convergence criterion of Pringsheim and even after the above conclusion II converges just as the regular continued fraction
Here is
Wherein represents a constant which is related to the Euler Gompertz constants. As Carl Ludwig Siegel has shown, also belongs to the transcendental numbers. So there is also here that the number is transcendental.
Example III
After the conclusion II above eventually converges for arbitrary, always the following infinite continued fraction:
The following applies:
Particular, it follows for:
And so
Example IV
Substituting in Example III, which also gives a convergent infinite continued fraction, in which case the convergence is not ensured by the convergence criterion of Pringsheim, but by the Seidel- the Stern criterion.
It is namely
Where is the golden number.
Counterexample
Is set in Example III, ie equal to the imaginary unit, we get no convergent infinite continued fraction. The infinite continued fraction
Is so divergent, although the number
With itself also diverges.
This shows that the set of Star pride generally specifies only a necessary but not a sufficient condition for the convergence of infinite regular continued fractions.
Application: representation of real numbers by negative - regular continued fractions
An infinite continued fraction of the real form
To natural numbers and integer initial term is called negative - regularly by Alfred Pringsheim.
The name is explained by the close relationship with the regular continued fractions, which Pringsheim also covered in his lectures on arithmetic and function theory.
Each infinite negative - regular continued fraction is convergent after pringsheimschen convergence criterion.
On this basis, we obtain the following representation theorem:
Formulation of the representation theorem
The amount of the infinite negative - regular continued fractions and the set of real numbers are in bijection with each other in such a way that each real number by an infinite negative - regular continued fraction of the form ( *) is displayed, the sequence of partial denominator is determined by clearly.
Addition I: Algorithm for determining the partial denominators
The partial denominators can be obtained by the following algorithm:
Is for general
The smallest integer greater. It has therefore always
And thus using the Gaußklammerfunktion
Consequently, always
Thus a sequence is first defined by recursion:
Then added
Addition II: distinction of rational and irrational numbers
A rational number is characterized in that in their representation (*) after a certain index for each partial denominator, while an irrational number is characterized in that in their representation (*) are infinitely many partial denominators.
Examples of negative - regular continued fraction representations
The following examples can be stated here:
This follows because directly from Part III of the pringsheimschen criterion.