The Bernoulli numbers or Bernoulli numbers, 1, ± 1 ⁄ 2 , 1 ⁄ 6 , 0, - 1 ⁄ 30 , ... are a sequence of rational numbers that occur in mathematics in different contexts: in the expansion coefficients trigonometric , hyperbolic and other functions , in the Euler-Maclaurin formula and in number theory in connection with the Riemann zeta function . The naming of these numbers after their discoverer Jakob I Bernoulli was introduced by Abraham de Moivre .
In the mathematical specialist literature, the Bernoulli numbers are defined as three different sequences that are very closely related. There is the older notation (mainly used until the 20th century), which is denoted here with , and the two newer forms, which in this article are denoted with and and have mostly been used since the middle of the 20th century. A more precise distribution or the historical transition of the conventions is difficult to objectify, since this strongly depends on the mathematician and the area of distribution of his writings. An implicit definition of Bernoulli numbers that is common nowadays is to use the coefficients of the following Taylor series either as
or (by mirroring on the y-axis) as
or earlier than
to introduce. Here are the numbers and the coefficients of the series expansion or the terms of the Bernoulli number sequence. The series expansions converge for all x with Replaced by , one recognizes the validity of , i. H. the first two definitions only differ for index 1 , all others or with an odd index are zero. To distinguish clearly, the links can be referred to as those of the first type (with ) and those of the second type (with ).
The older definition is based on the last row listed; in this only terms with indices occur, i. H. the terms with index 0 and 1 must be considered separately. For the remaining coefficients with an even index (precisely these are not zero) one chooses a separate definition so that they are all positive. Therefore applies
This is exactly what Jakob I Bernoulli had done when he was first identified and thus established the older notation, although he had not yet numbered it. He discovered these numbers by considering the polynomials, which describe the sum of the powers of natural numbers from 1 to a given one with small integer exponents. E.g.
This ultimately leads via Faulhaber's formulas to the Euler-Maclaurin formula , in which the Bernoulli numbers play a central role. He did not prove their general values, only correctly calculated those of the smaller coefficients - his corresponding notes were published posthumously .
The first Bernoulli numbers , ≠ 0 are
||to 6 decimal places
The numbers form a strictly convex (their differences grow) sequence. The denominators are always a multiple of 6, because
the Clausen and von-Staudt theorem , also known as Staudt-Clausen's theorem , applies :
It is named after the independent discovery by Thomas Clausen and Karl von Staudt in 1840. The denominator is the product of all prime numbers for which it is true that the index divides. Using Fermat's Little Theorem it follows that the factor converts these rational numbers into whole numbers.
Even if the result of the first magnitude relatively small number of values assumes goes with increasing but more quickly to infinity than any exponential . So is z. B.
Your asymptotic behavior can be compared with
describe, therefore the radius of convergence of the Taylor series, which were used for their definition above, is the same
If one would like to describe the Bernoulli numbers of the first kind , then these Bernoulli numbers result from the recursion formula with
and the start value . For odd indices it follows again . This formula comes from the implicit definition of the Bernoulli numbers of the first kind, which was also the most common definition until the middle of the 20th century because it has an easy-to-remember form:
which can also be written in the less common form than
In these representations, powers of are to be interpreted as the correspondingly indexed Bernoulli numbers. For the Bernoulli numbers of the second kind one can use analogously
as well as
or more elegant
write and use as an inductive definition of the Bernoulli numbers of the second kind with to
with the start value or for all as
Rows with Bernoulli numbers
These numbers occur, for example, in the Taylor series of the tangent , the hyperbolic tangent or the cosecan ; in general, when a function has a closed representation where the sine function (or hyperbolic sine function) is in the denominator - i.e. H. is divided by the sum or difference of two exponential functions :
Here are two non-converging asymptotic series that correspond to the trigamma function (the second derivative of the natural logarithm of the gamma function )
and that of the natural logarithm of the gamma function
which is known as the logarithm of the Stirling formula. This can be easily derived from the asymptotic form of the Euler-Maclaurin formula , in its symmetrical notation
is - in which case the term that (especially for th derivation of the integral) of the function evaluated at the location means - when there is, the lower summation limit to selected and the upper summation limit with holding variable. This is one of the best known applications of Bernoulli numbers and applies to all analytic functions , even if this asymptotic expansion does not converge in most cases.
Connection with the Riemann zeta function
The following series developments provide the (in the above sense) "classic" Bernoulli numbers:
The following applies to the “modern” Bernoulli numbers
wherein in the case of the more recent definition for n = 1 undefined terms of shape occur, but according to the rule of de l'Hospital because of the first-order pole of the Riemann zeta function at 1 (or in the last representation the term cancel in the denominator), and thus correctly deliver the value .
For the Bernoulli numbers of the second kind there is also the concise representation
so that the entire theory of the Riemann zeta function for characterizing the Bernoulli numbers is available.
For example, the product representation of the Riemann zeta function and the above series expansion of the Bernoulli numbers result in the following representation:
Here the product extends over all prime numbers (see also Euler product of the Riemann zeta function).
There are many improper integrals with sums or differences of two exponential functions in the denominator of the integrand whose values are given by Bernoulli numbers. Some simple examples are
The graphs of the Bernoulli polynomials of degrees 1 to 6
The Bernoulli polynomial is a mapping for each and is fully characterized by the following recursion equations: For we set
and for the -th Bernoulli polynomial results uniquely from the two conditions
recursively from the previous one. As the sum of the powers of written, the expression for the -th polynomial is
where here again denote the Bernoulli numbers of the first kind. This form follows directly from the symbolic formula
in which the powers of are interpreted as the corresponding nth Bernoulli number . The first Bernoulli polynomials are
These polynomials are symmetrical around , more precisely
Their constant terms are the Bernoulli numbers of the first kind, that is
the Bernoulli numbers of the second kind are obtained from
and finally applies
in the middle of the interval. The kth Bernoulli polynomial has fewer than k zeros in the whole interval for k> 5 and two for even n ≠ 0 and the three zeros for odd n ≠ 1 in the interval . Let be the set of zeros of these polynomials. Then
for all n ≠ 5 and n ≠ 2 and it holds
where the function applied to a set indicates the number of elements.
The function values of the Bernoulli polynomials in the interval [0,1] are through for an even index
and for odd index (but not sharp) through
They also satisfy the equation
if they are continued analytically and the sum of the power of the first n natural numbers can be expressed as
describe. The index shift from to on the right-hand side of the equation is necessary here because historically the Bernoulli poynomials were "incorrectly" attached to the Bernoulli numbers of the first type (and not the second type) and thus obtained instead of the summands in the above Bernoulli poynomials, what exactly results in the value too little here (the last term of the sum on the left side) and therefore this index has to run “one further” on the right side.
Bernoulli numbers in algebraic number theory
As a sentence from Staudt-Clausen is also the statement
known, which is somewhat stronger than the previous theorem by Clausen and von Staudt for characterizing the denominator. The sequence of the whole numbers determined in this way for an even index is .
An odd number is called regular prime , if they do not counter the Bernoulli numbers with shares. Kummer showed that this condition is equivalent to not dividing the class number of the p-th pitch field . In 1850 he was able to prove that Fermat's great theorem , namely has for no solutions in , holds for all exponents that are a regular prime number. For example, by checking the Bernoulli numbers up to index 94, Fermat's great theorem was proven for all other exponents with the exception of the exponents 37, 59, 67 and 74 .
Tangent numbers and applications in combinatorics
If one considers the Euler's numbers and the Taylor expansion of the tangent function, then one can define the tangent numbers implicitly
and still set for index zero . So you have the transformation
which generates this sequence of integers from the Bernoulli numbers of the first kind:
Since the choice of sign in the implicit definition is completely arbitrary, one can just as justifiably use
define the tangent numbers, with the consequence
and has indexes for all
In any case, with the exception of, all numbers with an even index are zero and those with an odd index have alternating signs.
The values are now exactly the number of alternating permutations of an elementary set. Further information on the direct determination of tangent numbers can be found in the article Euler's numbers .
In combinatorics, the Bernoulli numbers of the second kind can also be represented by the Stirling numbers of the second kind as
The values are also known as Worpitzky numbers . Another connection arises from the generating power series of the Stirling polynomials with paths
with the Stirling numbers of the first kind to
which one could define as negative . Hence the Bernoulli numbers of the second kind are also the values of the sterling polynomials at zero
due to the same formal power series.
Here in the article, the Bernoulli numbers were initially defined arbitrarily by means of generating power series. The formal power series of also occurs directly when determining the Todd class of a vector bundle in a topological space :
where are the cohomology classes of . If is finite dimensional, then is a polynomial. The Bernoulli numbers of the second kind naturally "count" certain topological objects. This formal power series is also reflected in the L gender or Todd gender of the characteristic power series of an orientable manifold .
Jakob Bernoulli : Ars conjectandi , opus posthumum. (Art of guessing, work left behind), Basileæ (Basel) 1713 (Latin).
Julius Worpitzky : Studies on Bernoulli and Euler's numbers . Crelles Journal 94, 1883, pp. 203-232.
Senon I. Borewicz , Igor R. Šafarevič : Number theory. Birkhäuser Verlag Basel, 1966, chap. 5, § 8, pp. 408-414.
Jürgen Neukirch : Algebraic number theory. Springer-Verlag, 1992.
- Kenneth F. Ireland, Michael Rosen : A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, 2nd edition 1990, chap. 15, pp. 228-248.
IS Gradshteyn, IM Ryzhik : Table of Integrals, Series and Products. Academic Press, 4th edition 1980, ISBN 0-12-294760-6 , chap. 9.6.
- Ulrich Warnecke: For the polynomial representation of for any In: Mathematical semester reports. Volume XXX / 1983, pp. 106-114.
^ JC Kluyver: The Staudt-Clausen'sche theorem. Math. Ann. Vol. 53, (1900), pp. 591-592.
^ W. Gröbner and N. Hofreiter: Integraltafel . Second part: definite integrals. 5. verb. Edition, Springer-Verlag, 1973.
^ John H. Conway , Richard K. Guy : The Book of Numbers. Springer-Verlag, 1996, ISBN 0-387-97993-X , chap. 4, pp. 107-109.
↑ JM Borwein , PB Borwein , K. Dilcher: Pi, Euler Numbers, and Asymptotic Expansions. AMM, Vol. 96, No. 8, (Oct. 1989), p. 682.
^ Henry Wadsworth Gould: Combinatorial identities. Morgantown, W Va, 1972.
↑ K. Reillag, J. Gallier: Complex Algebraic Geometry. CIS 610, Lecture Notes, Fall 2003 - Spring 2004, Chap 3, pp. 209–220 (online).