Pringsheim's convergence criterion

The Pringsheim convergence criterion or also the main Pringsheim criterion is a criterion about the convergence behavior of infinite continued fractions . It goes back to the German mathematician Alfred Pringsheim and is one of the classic tenets of continued fraction theory within analytical number theory . In the English-language specialist literature , the criterion is also referred to as Śleszyński-Pringsheim's theorem (etc.), the first name referring to the Polish-Russian mathematician Ivan Śleszyński (1854–1931), who also used this criterion before Pringsheim had found. There is evidence that Alfred Pringsheim may have been aware of the corresponding publication by Ivan Śleszyński when he made his publication in 1898. However, Oskar Perron's reference in Volume II of his theory of continued fractions should also be added , according to which the essential content of this sentence can already be found in the textbook on algebraic analysis by Moritz Abraham Stern (Leipzig 1860).

Formulation of the criterion

part One

For two sequences of numbers of complex numbers     and     with the property that the inequalities ${\displaystyle (a_{i})_{i=1,2,3\dots }}$ ${\displaystyle (b_{i})_{i=1,2,3\dots }}$

${\displaystyle |b_{i}|\geq |a_{i}|+1}$   ${\displaystyle (i=1,2,3\dots )}$

are fulfilled, is the associated continued fraction

${\displaystyle {\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {a_{i}}{b_{i}}}={\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\ddots }}}}}}}$

always convergent. That means:

The consequence of the approximation breaks

${\displaystyle f_{n}={\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{\ddots +{\cfrac {a_{n}}{b_{n}}}}}}}}}}$   ${\displaystyle (n=1,2,3,\dots )}$

is a convergent sequence and the limit value clearly determined by it with ${\displaystyle f\in \mathbb {C} }$

  ${\displaystyle f=\lim _{n\to \infty }f_{n}={\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\ddots }}}}}}}$  .

is the value of the associated continued fraction .

Part II

In the event that the above condition is met, this always applies

${\displaystyle |f_{n}|<1}$   ${\displaystyle (n=1,2,3\dots )}$   and with it   . ${\displaystyle |f|\leq 1}$

Part III

The borderline case     exists if and only if the following three conditions are met: ${\displaystyle |f|=1}$

( IIIa )   ${\displaystyle |b_{i}|={|a_{i}|+1}}$   ${\displaystyle (i=1,2,3\dots )}$

( IIIb ) All   are negative real numbers . ${\displaystyle {\frac {a_{i+1}}{{b_{i}}\cdot {b_{i+1}}}}}$   ${\displaystyle (i=1,2,3\dots )}$

( IIIc ) The series     is divergent . ${\displaystyle {\sum _{i=1}^{\infty }{|a_{1}\cdot a_{2}\cdots a_{i}|}}}$

In this borderline case , the continued fraction has the value  ${\displaystyle f={\frac {a_{1}\cdot |b_{1}|}{|a_{1}|\cdot b_{1}}}}$

Inferences

From the convergence criterion of Pringsheim several other convergence criteria can be derived. These include the following:

Corollary I: Worpitzky's theorem

For a number sequence of complex numbers    , which has the inequality ${\displaystyle (a_{i})_{i=1,2,3\dots }}$

${\displaystyle |a_{i}|\leq {\frac {1}{4}}}$   ${\displaystyle (i=1,2,3\dots )}$

fulfilled, is the continued fraction

${\displaystyle {\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {a_{i}}{1}}={\cfrac {a_{1}}{1+{\cfrac {a_{2}}{1+{\cfrac {a_{3}}{1+\ddots }}}}}}}$

always convergent.

The following   always applies to the approximate fractions   ${\displaystyle f_{n}}$   ${\displaystyle (i=1,2,3\dots )}$

${\displaystyle |f_{n}|<{\frac {1}{2}}}$

and accordingly for the value of     the continued fraction ${\displaystyle f}$

${\displaystyle |f|\leq {\frac {1}{2}}}$

The set of Worpitzky was in 1865 by Julius Worpitzky published and is considered the first convergence criterion for continued fractions with elements of the complex plane .

Corollary II: Another convergence criterion from Pringsheim

Through specialization, one finds another criterion of convergence by Pringsheim , which Alfred Pringsheim himself formulated in his work On the Convergence of Infinite Continued Fractions in the meeting reports of the Bavarian Academy of Sciences from 1898 and which reads as follows:

For a number sequence of complex numbers    , which has the inequality ${\displaystyle (b_{i})_{i=1,2,3\dots }}$

${\displaystyle {{\frac {1}{|b_{2i-1}|}}+{\frac {1}{|b_{2i}|}}}\leq 1}$   ${\displaystyle (i=1,2,3\dots )}$

is fulfilled, is the regular continued fraction

${\displaystyle {\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {1}{b_{i}}}={\cfrac {1}{b_{1}+{\cfrac {1}{b_{2}+{\cfrac {1}{b_{3}+\ddots }}}}}}}$

always convergent.

This further convergence criterion from Pringsheim is always applicable, for example, in the event that all participant denominators have   at least the amount 2. ${\displaystyle b_{i}}$   ${\displaystyle (i=1,2,3\dots )}$

Associated criteria: Stern-Stolz's and Seidel-Stern's theorems and Tietze's convergence theorem

In the case of regular infinite continued fractions , there are some criteria with regard to the question of convergence and divergence , which are repeatedly used as a supplement to Pringsheim's convergence criterion . These include the following theorems, which, in addition to this, are among the classic results of continued fraction convergence theory.

Set of star pride

The set of Star Pride formulated a very common condition for the divergence regular infinite continued fractions and reads as follows:

Any complex continued fraction

${\displaystyle {{\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {1}{b_{i}}}}}$

to a sequence of complex numbers   ${\displaystyle b_{i}\in \mathbb {C} }$   ${\displaystyle (i=1,2,3\dots )}$  

is divergent in any case if the associated series

${\displaystyle \sum _{i=1}^{\infty }b_{i}}$

is absolutely convergent . That means: For the convergence of the continued fraction it is always necessary that

${\displaystyle \sum _{i=1}^{\infty }{|b_{i}|}=\infty }$

applies.

This criterion goes back to Moritz Abraham Stern and Otto Stolz .

Set of Seidel-Stern

The set of Seidel Star exacerbated the set of Star Pride in case regular infinite continued fractions with consistently positive part denominators by even the latter condition as a necessary and sufficient condition identifies. So it reads:

The continued fraction   converges for a sequence of positive real numbers   ${\displaystyle (b_{i})_{i=1,2,3\dots }}$

${\displaystyle {{\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {1}{b_{i}}}}}$

then and only if the associated series

${\displaystyle \sum _{i=1}^{\infty }b_{i}}$

diverges.

This criterion goes back to Philipp Ludwig von Seidel and Moritz Abraham Stern. It takes effect when the in Part I of pringsheimschen criterion inequality above is not consistently achievable, however, in connection with the assumed positivity of the partial denominator can be replaced by the series divergence condition.

Tietze's convergence theorem

The convergence theorem of Tietze also deals with the convergence of infinite continued fractions . It goes back to the German mathematician Heinrich Tietze and says the following:

Let there be two sequences of real numbers     and     that   satisfy the following three conditions for all indices : ${\displaystyle (a_{i})_{i=1,2,3\dots }}$ ${\displaystyle (b_{i})_{i=0,1,2,3\dots }}$   ${\displaystyle i\in \mathbb {N} }$

  (I)   ${\displaystyle |a_{i}|=1}$

  (II)   ${\displaystyle b_{i}\geq 1}$

  (III)   ${\displaystyle b_{i}+a_{i+1}\geq 1}$

Then is the associated continued fraction

  (*)   ${\displaystyle b_{0}+{\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {a_{i}}{b_{i}}}}$

always convergent. The consequence of the approximation breaks

${\displaystyle f_{n}=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{\ddots +{\cfrac {a_{n}}{b_{n}}}}}}}}}}$   ${\displaystyle (n=0,1,2,3,\dots )}$

converges to     the limit value ${\displaystyle \mathbb {R} }$

${\displaystyle f=\lim _{n\to \infty }f_{n}=b_{0}+{\cfrac {a_{1}}{b_{1}+{\cfrac {a_{2}}{b_{2}+{\cfrac {a_{3}}{b_{3}+\ddots }}}}}}}$

and it applies

${\displaystyle b_{0}<f\leq b_{0}+1}$if   , ${\displaystyle a_{1}=1}$

or.

${\displaystyle b_{0}-1\leq f<b_{0}}$if     . ${\displaystyle a_{1}=-1}$

In addition, the denominators of   the approximate fractions     always satisfy   the inequality ${\displaystyle B_{n}}$ ${\displaystyle f_{n}}$     ${\displaystyle (n=0,1,2,3\dots )}$

${\displaystyle B_{n}\geq 1}$

and it is

${\displaystyle \lim _{n\to \infty }B_{n}=\infty }$

Connection with irrationality

Starting with Tietze's convergence theorem, statements about irrationality can be achieved. As Heinrich Tietze himself proved, every infinite continued fraction of the form (*)   always converges   - with one single exception! - against an irrational number   , provided that the conditions are tightened as follows: ${\displaystyle f}$

  (Ia)   ${\displaystyle |a_{i}|=1}$

  (IIb)    ,   (IIa) ${\displaystyle b_{0}\in \mathbb {Z} }$   ${\displaystyle b_{i}\in \mathbb {N} }$

  (IIIa)    , provided   ${\displaystyle a_{i+1}=1}$ ${\displaystyle b_{i}=1}$

  ${\displaystyle (i=1,2,3\dots )}$

The exception is when   the following exception condition   (A) is also   met   for all indices   from one index   : ${\displaystyle i_{0}}$ ${\displaystyle i>i_{0}}$

  (A)   ,   ${\displaystyle a_{i}=-1}$ ${\displaystyle b_{i}=2}$

In this exceptional case, the limit value is     a rational number . ${\displaystyle f}$

Examples and application

Example I.

According to the Pringsheim convergence criterion , the following infinite continued fraction converges:

${\displaystyle {\begin{aligned}g&={\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {i}{i+1}}\\&={\cfrac {1}{2+{\cfrac {2}{3+{\cfrac {3}{4+{\cfrac {4}{\;\,\ddots }}}}}}}}\\\end{aligned}}}$

Since ( IIIb ) is not fulfilled,   part III is   not applicable. Rather is

${\displaystyle g={\frac {1}{e-2}}-1=0{,}3922111911773330\dotso }$  ,

as can be seen from the continued fraction expansions of Euler's number found by Leonhard Euler and Ernesto Cesàro   . Therefore, because of the transcendence of Euler's number, the number is     also a transcendent number .   ${\displaystyle e}$ ${\displaystyle g}$

Example II

According to Pringsheim's convergence criterion and even according to Corollary II mentioned above, the regular continued fraction converges in the same way

${\displaystyle {\begin{aligned}h&={\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {1}{i+1}}\\&={\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {1}{4+{\cfrac {1}{\;\,\ddots }}}}}}}}\\\end{aligned}}}$ .

Here is

${\displaystyle h={\frac {1}{c_{1}}}-1=0{,}433127426\dotso }$  ,

where represents     a constant which is related to the so-called Euler-Gompertz constant . As Carl Ludwig Siegel has shown, is also     one of the transcendent numbers. So here, too, the number is     transcendent. ${\displaystyle c_{1}=0{,}6977746579\dotso }$ ${\displaystyle c_{1}}$ ${\displaystyle h}$

Example III

According to Corollary II mentioned above ,     the following infinite continued fraction also converges for arbitrary things: ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle |z|\geq 2}$

${\displaystyle {\begin{aligned}f(z)&={\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {1}{z}}\\&={\cfrac {1}{z+{\cfrac {1}{z+{\cfrac {1}{z+\ddots }}}}}}\\\end{aligned}}}$

The following applies here:

${\displaystyle {\begin{aligned}f(z)&={\frac {1}{z+f(z)}}\\&={\frac {{\sqrt {z^{2}+4}}-z}{2}}\\\end{aligned}}}$   .

In particular    : ${\displaystyle z=2}$

${\displaystyle {\begin{aligned}f(2)&={\frac {{\sqrt {8}}-2}{2}}\\&={\sqrt {2}}-1\\\end{aligned}}}$  

and so

${\displaystyle {\begin{aligned}{\sqrt {2}}&=1+{{\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {1}{2}}}\\&=1+{\cfrac {1}{2+{\cfrac {1}{2+{\cfrac {1}{2+\ddots }}}}}}\\\end{aligned}}}$   .

Example IV

If you start in Example III     , you also get a convergent infinite continued fraction   , although here the convergence is not secured by the Pringsheim convergence criterion , but by the Seidel-Stern criterion . ${\displaystyle z=1}$ ${\displaystyle \psi }$

It is true

${\displaystyle {\begin{aligned}\psi &={{\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {1}{1}}}\\&=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+\ddots }}}}}}\\&={\frac {{\sqrt {5}}-1}{2}}\\&={\frac {1}{\Phi }}\\\end{aligned}}}$,

where stands     for the golden number . ${\displaystyle \Phi }$

Counterexample

If in example III is     set, i.e. equal to the imaginary unit , then one does not get a convergent infinite continued fraction. The infinite chain fraction ${\displaystyle z={\mathrm {i} }}$

${\displaystyle {\begin{aligned}f(z)&={\underset {n=1}{\overset {\infty }{\mathbf {K} }}}{\frac {1}{\mathrm {i} }}\\&={\cfrac {1}{{\mathrm {i} }+{\cfrac {1}{{\mathrm {i} }+{\cfrac {1}{{\mathrm {i} }+\ddots }}}}}}\\\end{aligned}}}$

is therefore divergent, although the series

${\displaystyle \sum _{n=1}^{\infty }b_{n}}$

with     itself also diverges. ${\displaystyle b_{n}={\mathrm {i} }}$   ${\displaystyle (n=1,2,3\dots )}$

This shows that Stern-Pride's theorem gives in general only one necessary but not a sufficient condition for the convergence of regular infinite continued fractions.

Application: Representation of real numbers by negative-regular continued fractions

An infinite real continued fraction of form

(*) ${\displaystyle b_{0}+{{\underset {i=1}{\overset {\infty }{\mathbf {K} }}}{\frac {-1}{b_{i}}}}=b_{0}-{\cfrac {1}{b_{1}-{\cfrac {1}{b_{2}-{\cfrac {1}{b_{3}-\ddots }}}}}}}$

To natural numbers       with     and to an integer initial term     is called negative-regular according to Alfred Pringsheim . ${\displaystyle b_{i}\in \mathbb {N} }$ ${\displaystyle b_{i}\geq 2}$   ${\displaystyle (i=1,2,3\dots )}$ ${\displaystyle b_{0}\in \mathbb {Z} }$

The naming is explained by the close relationship with the regular continued fractions , which Pringsheim also deals with in his lectures on number and function theory .

Every infinite negative-regular continued fraction is convergent according to Pringsheim's convergence criterion.

On the basis of this, the following notation is obtained :

Formulation of the notation

The set of infinite negative-regular continued fractions and the set of real numbers are bijectioned to one another in such a way that each real number can be represented     by an infinite negative-regular continued fraction of the form (*) , whereby the sequence of the denominators   is     clearly determined by . ${\displaystyle \xi \in \mathbb {R} }$   ${\displaystyle (b_{i})_{i=0,1,2,3\dots }}$ ${\displaystyle \xi }$

Addition I: Algorithm for determining the part denominators

The part denominators can be obtained using the following algorithm :

For general     is ${\displaystyle x\in \mathbb {R} }$

${\displaystyle G(x)}$  

the smallest whole number is greater   ${\displaystyle x}$   . So you always have

${\displaystyle x<G(x)\leq x+1}$  

and thus using the Gaussian bracket function

${\displaystyle G(x)={\lfloor x\rfloor }+1}$   .

Hence is always

${\displaystyle {\frac {1}{G(x)-x}}\geq 1}$   .

First, a sequence is     defined using recursion : ${\displaystyle (x_{i})_{i=0,1,2,3\dots }}$

${\displaystyle x_{0}:=\xi }$

${\displaystyle x_{i}:={\frac {1}{G(x_{i-1})-x_{i-1}}}}$   ${\displaystyle (i=1,2,3\dots )}$  

Then you bet

${\displaystyle b_{i}:=G(x_{i})}$   ${\displaystyle (i=0,1,2,3\dots )}$  .

Addition II: Distinction between rational and irrational numbers

A rational number     is characterized in that in their representation (*) after a certain index   for       each part of the denominator   is, while an irrational number   is characterized in that in their representation (*) infinite number of denominators   are   . ${\displaystyle \xi \in \mathbb {Q} }$ ${\displaystyle i_{\xi }\in \{0,1,2,3\dots \}}$ ${\displaystyle i\geq i_{\xi }}$   ${\displaystyle b_{i}=2}$   ${\displaystyle \xi \in {\mathbb {R} \setminus \mathbb {Q} }}$   ${\displaystyle b_{i_{k}}\geq 3}$ ${\displaystyle (k=0,1,2,3\dots )}$

Examples of negative-regular continued fraction representations

The following examples can be given:

1. Representation of the 1

${\displaystyle 1=2-{\cfrac {1}{2-{\cfrac {1}{2-{\cfrac {1}{2-{\cfrac {1}{2-{\cfrac {1}{2-{\cfrac {1}{2-{\cfrac {1}{2-{\cfrac {1}{2-\dotsb }}}}}}}}}}}}}}}}}$

This follows directly from Part III of the Pringsheim criterion . ${\displaystyle -1=1-2}$

2. Representation of the root of 2

${\displaystyle {\sqrt {2}}=2-{\cfrac {1}{2-{\cfrac {1}{4-{\cfrac {1}{2-{\cfrac {1}{4-{\cfrac {1}{2-{\cfrac {1}{4-{\cfrac {1}{2-{\cfrac {1}{4-\dotsb }}}}}}}}}}}}}}}}}$

3. Representation of the root of 3

${\displaystyle {\sqrt {3}}=2-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-\dotsb }}}}}}}}}}}}}}}}}$

4. Representation of the root of 7

${\displaystyle {\sqrt {7}}=3-{\cfrac {1}{3-{\cfrac {1}{6-{\cfrac {1}{3-{\cfrac {1}{6-{\cfrac {1}{3-{\cfrac {1}{6-{\cfrac {1}{3-{\cfrac {1}{6-\dotsb }}}}}}}}}}}}}}}}}$

5. Representations of the golden number

(a) ${\displaystyle {\begin{aligned}{\Phi }&={\frac {{\sqrt {5}}+1}{2}}\\&=2-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-\dotsb }}}}}}}}}}}}}}}}\\\end{aligned}}}$  

(b) ${\displaystyle {\begin{aligned}{\frac {1}{\Phi }}&={\Phi }-1\\&={\frac {{\sqrt {5}}-1}{2}}\\&=1-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-{\cfrac {1}{3-\dotsb }}}}}}}}}}}}}}}}\\\end{aligned}}}$  

Remarks

1. Further convergence criteria for infinite continued fractions go back to Alfred Pringsheim. There are also a significant number of other convergence criteria.
2. From the representation theorem it follows immediately that the set of real numbers is of uncountable power .

literature

• Steven R. Finch: Mathematical Constants (=  Encyclopedia of Mathematics and its Applications . Volume 94 ). Cambridge Univity Press, Cambridge et al. 2003, ISBN 0-521-81805-2 .
• Lisa Lorentzen, Haakon Waadeland: Continued Fractions with Applications (=  Studies in computational mathematics . Volume 3 ). Elsevier, Amsterdam et al. 1992, ISBN 0-444-89265-6 .
• William B. Jones, WJ Throne: Continued Fractions. Analytic Theory and Applications (=  Encyclopedia of Mathematics and its Applications . Volume 11 ). Addison-Wesley Publishing Company, Reading, Mass. et al. 1980, ISBN 0-201-13510-8 .
• Oskar Perron: The theory of continued fractions . Volume I: Elementary Continued Fractions. . Teubner Verlag, Stuttgart 1977, ISBN 3-519-02021-1 (Reprographic reprint of the 3rd, improved and expanded edition, Stuttgart 1954).
• Oskar Perron: The theory of continued fractions . Volume II: Analytical-function-theoretical continued fractions. . Teubner Verlag, Stuttgart 1977, ISBN 3-519-02022-X (Reprographic reprint of the 3rd, improved and expanded edition, Stuttgart 1957).
• Alfred Pringsheim: About the convergence of infinite continued fractions . In: Meeting reports of the (royal) Bavarian Academy of Sciences in Munich. Mathematical-physical (natural science) class . tape 28 , 1898, pp. 295-324 ( zbmath.org ).
• Alfred Pringsheim: Lectures on numbers and functions. First volume: theory of numbers. Third section: Complex numbers - series with complex members - infinite products and continued fractions (=  BG Teubner's collection of textbooks in the field of mathematical sciences including their applications . XL, I.3). Teubner Verlag, Leipzig and Berlin 1921.
• WJ Throne: Should the Pringsheim criterion be renamed the Śleszyński criterion? In: Comm. Anal. Theory Contin. Fractions . tape 1 , 1992, p. 13-20 ( MR1192192 ).
• Heinrich Tietze: On criteria for convergence and irrationality of infinite continued fractions . In: Math. Ann. tape 70 , 1911, pp. 236-265 ( digizeitschriften.de ).
• Alfred Pringsheim: About the convergence of infinite continued fractions . In: Meeting reports of the (royal) Bavarian Academy of Sciences in Munich. Mathematical-physical (natural science) class . tape 28 , 1898, pp. 295-324 ( zbmath.org ).
• Wacław Sierpiński : Elementary Theory of Numbers (=  North-Holland Mathematical Library . Volume 31 ). 2nd revised and expanded edition. Elsevier, Amsterdam (et al.) 1988, ISBN 0-444-86662-0 .
• Hubert Stanley Wall: Analytic Theory of Continued Fractions. (=  Chelsea Scientific Books . Volume 207 ). Chelsea Publishing Company, Bronx, NY 1967, ISBN 0-8284-0207-8 (reprint of Van Norstrand, New York 1948 edition).
• J. Worpitzky : Investigations into the development of the monodromic and monogenic functions through continued fractions . In: Friedrichs-Gymnasium and Realschule: annual report . 1865, p. 3-39 .

References and footnotes

1. ↑ Platform: p. 58.
2. ^ Pringsheim: Lectures ... Volume I.3 , p. 878 ff .
3. ↑ Lorentzen, Waadeland: p. 30 ff.
4. ↑ Throne: Should the Pringsheim criterion be renamed the Śleszyński criterion? In: Comm. Anal. Theory Contin. Fractions . tape 1 , 1992, p. 13 ff .
5. ↑ Since the initial link never has an influence on the convergence and divergence of the continued fractions, it is usually not mentioned in the following when formulating the convergence criteria. By adding an initial link, the convergence and divergence of a continued fraction always remain unaffected. ${\displaystyle b_{0}}$
6. ↑ stands for the complex amount . ${\displaystyle |\cdot |}$
7. ^ Platform: pp. 61–62.
8. ↑ Lorentzen, Waadeland: p. 135.
9. ^ Jones, Throne: 94.
10. ^ Worpitzky: Investigations ... In: Annual report . 1865, p. 29-30 .
11. ↑ Jones, Throne: pp. 10, 94.
12. ↑ It is also mentioned in his lectures on numbers and functions ; s. Volume I.3, p. 880.
13. ↑ ^{a b} platform: p. 42.
14. ↑ Lorentzen, Waadeland: p. 94.
15. ^ Pringsheim: Lectures ... Volume I.3 , p. 846 .
16. ^ Jones, Throne: 79.
17. ↑ Lorentzen, Waadeland: p. 94.
18. ^ Pringsheim: Lectures ... Volume I.3 , p. 846, 966 .
19. ↑ However, in connection with this sentence in HS Wall: pp. 27–28, 424. to Helge von Koch and his work Sur un théorème de Stieltjes et sur les fractions continues . In: Bull. Soc. Math. De France . tape 23 , 1895, p. 23-40 . referred!
20. ↑ Platform: p. 46.
21. ↑ Lorentzen, Waadeland: p. 98.
22. ^ Pringsheim: Lectures ... Volume I.3 , p. 764, 962 .
23. ↑ In Jones, Thron: p. 87. Seidel-Stern's theorem is presented in a somewhat tightened version, which includes statements about the convergence behavior of the approximate fractions.
24. ^ Platform: p. 135 ff.
25. ↑ Tietze: On criteria for convergence and irrationality of infinite continued fractions . In: Math. Ann. tape 70 , 1911, pp. 236 ff .
26. ↑ stands for the amount function . ${\displaystyle |\cdot |}$
27. ↑ Tietze: On criteria for convergence and irrationality of infinite continued fractions . In: Math. Ann. tape 70 , 1911, pp. 246 ff .
28. ↑ Platform: p. 19.
29. ↑ See Finch: p. 423.
30. ↑ Lorentzen, Waadeland: p. 32.
31. ↑ This is the main value of the complex square root function .
32. ↑ Lorentzen, Waadeland: p. 46.
33. ^ Wall: p. 29.
34. ↑ The regular continued fractions are characterized by the fact that they regularly are that all their denominators from index 1 are natural numbers and that the initial term of each integer is. The difference between negative-regular continued fractions and regular continued fractions is therefore the sign of the partial counter and the fact that in the regular continued fractions is also approved the partial denominator first Both finite and infinite continued fractions are considered in both cases. Here only the infinite continued fractions play a role. See Pringsheim: Lectures ... Volume I.3 , p. 752 ff., 773 ff., 812 ff .
35. ↑ Likewise, every infinite regular continued fraction converges , namely according to Seidel-Stern's theorem; see. Pringsheim: lectures ... volume I.3 , p. 773 .
36. ^ ^{A b} Pringsheim: Lectures ... Volume I.3 , p. 819 .
37. ↑ ^{a b} Sierpiński: p. 337.
38. ^ Pringsheim: Lectures ... Volume I.3 , p. 818-819 .
39. ↑ Sierpiński: pp. 336–337.
40. ↑ Sierpiński: pp. 337–338.
41. ↑ Lorentzen, Waadeland: p. 562.
42. ↑ Platform: p. 38 ff.
43. ^ Jones, Throne: pp. 60-146.
44. ↑ Lorentzen, Waadeland: pp. 32–36.
45. ^ Wall: Analytic Theory…. Part I: Convergence Theory, p. 11-157 .