Mathematical constant

Constants in advanced mathematics

These are constants which are encountered frequently in higher mathematics.

The Euler–Mascheroni constant γ

Euler's constant or the Euler–Mascheroni constant is defined as the limiting difference between the harmonic series and the natural logarithm:

${\displaystyle \begin{array}{cc}\gamma & =\underset{n\to \mathrm{\infty }}{lim}\left(-\ln n+\sum _{k=1}^n\frac{1}{k}\right)\end{array}}$

It appears frequently in mathematics, especially in number theoretical contexts such as Mertens' third theorem or the growth rate of the divisor function. It has relations to the gamma function and its derivatives as well as the zeta function and there exist many different integrals and series involving ${\displaystyle \gamma }$.

Despite the ubiquity of the Euler-Mascheroni constant, many of its properties remain unknown. That includes the major open questions of whether it is a rational or irrational number and whether it is algebraic or transcendental. In fact, ${\displaystyle \gamma }$ has been described as a mathematical constant "shadowed only ${\displaystyle \pi }$ and ${\displaystyle e}$ in importance."

The numeric value of ${\displaystyle \gamma }$ is approximately:

0.57721566490153286060651209008240243104215933593992... (sequence A001620 in the OEIS).

Apéry's constant ζ(3)

Apery's constant is defined as the sum of the reciprocals of the cubes of the natural numbers:$${\displaystyle \zeta (3)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}=1+\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+\frac{1}{5^3}\cdots }$$It is the special value of the Riemann zeta function ${\displaystyle \zeta (s)}$ at ${\displaystyle s=3}$. The quest to find an exact value for this constant in terms of other known constants and elementary functions originated when Euler famously solved the Basel problem by giving ${\displaystyle \zeta (2)=\frac{1}{6}{\pi }^2}$. To date no such value has been found and it is conjectured that there is none. However, there exist many representations of ${\displaystyle \zeta (3)}$ in terms of infinite series.

Apéry's constant arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.

${\displaystyle \zeta (3)}$ is known to be an irrational number which was proven by the French mathematician Roger Apéry in 1979. It is however not known whether it is algebraic or transcendental.

The numeric value of Apéry's constant is approximately:

1.20205690315959428539973816151144999076498629234049... (sequence A002117 in the OEIS).

Catalan's constant G

Catalan's constant is defined by the alternating sum of the reciprocals of the odd square numbers:

${\displaystyle G=\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(2n+1{)}^2}=\frac{1}{1^2}-\frac{1}{3^2}+\frac{1}{5^2}-\frac{1}{7^2}+\frac{1}{9^2}-\cdots }$

It is the special value of the Dirichlet beta function ${\displaystyle \beta (s)}$ at ${\displaystyle s=2}$. Catalan's constant appears frequently in combinatorics and number theory and also outside mathematics such as in the calculation of the mass distribution of spiral galaxies.

Questions about the arithmetic nature of this constant also remain unanswered, ${\displaystyle G}$ having been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven." There exist many integral and series representations of Catalan's constant.

It is named after the French and Belgian mathematician Charles Eugène Catalan.

The numeric value of ${\displaystyle G}$ is approximately:

0.91596559417721901505460351493238411077414937428167... (sequence A006752 in the OEIS).

The Feigenbaum constants α and δ

Iterations of continuous maps serve as the simplest examples of models for dynamical systems. Named after mathematical physicist Mitchell Feigenbaum, the two Feigenbaum constants appear in such iterative processes: they are mathematical invariants of logistic maps with quadratic maximum points and their bifurcation diagrams. Specifically, the constant α is the ratio between the width of a tine and the width of one of its two subtines, and the constant δ is the limiting ratio of each bifurcation interval to the next between every period-doubling bifurcation.

The logistic map is a polynomial mapping, often cited as an archetypal example of how chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the Australian biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. The difference equation is intended to capture the two effects of reproduction and starvation.

The Feigenbaum constants in bifurcation theory are analogous to π in geometry and e in calculus. Neither of them is known to be irrational or even transcendental. However proofs of their universality exist.

The respective approximate numeric values of δ and α are:

4.66920160910299067185320382046620161725818557747576... (sequence A006890 in the OEIS).

2.50290787509589282228390287321821578638127137672714... (sequence A006891 in the OEIS).

Mathematical curiosities

Simple representatives of sets of numbers

Some constants, such as the square root of 2, Liouville's constant and Champernowne constant:

${\displaystyle C_{10}=0.12345678910111213141516\dots }$

are not important mathematical invariants but retain interest being simple representatives of special sets of numbers, the irrational numbers, the transcendental numbers and the normal numbers (in base 10) respectively. The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum who proved, most likely geometrically, the irrationality of the square root of 2. As for Liouville's constant, named after French mathematician Joseph Liouville, it was the first number to be proven transcendental.

Chaitin's constant Ω

In the computer science subfield of algorithmic information theory, Chaitin's constant is the real number representing the probability that a randomly chosen Turing machine will halt, formed from a construction due to Argentine-American mathematician and computer scientist Gregory Chaitin. Chaitin's constant, though not being computable, has been proven to be transcendental and normal. Chaitin's constant is not universal, depending heavily on the numerical encoding used for Turing machines; however, its interesting properties are independent of the encoding.

Notation

Representing constants

It is common to express the numerical value of a constant by giving its decimal representation (or just the first few digits of it). For two reasons this representation may cause problems. First, even though rational numbers all have a finite or ever-repeating decimal expansion, irrational numbers don't have such an expression making them impossible to completely describe in this manner. Also, the decimal expansion of a number is not necessarily unique. For example, the two representations 0.999... and 1 are equivalent in the sense that they represent the same number.

Calculating digits of the decimal expansion of constants has been a common enterprise for many centuries. For example, German mathematician Ludolph van Ceulen of the 16th century spent a major part of his life calculating the first 35 digits of pi. Using computers and supercomputers, some of the mathematical constants, including π, e, and the square root of 2, have been computed to more than one hundred billion digits. Fast algorithms have been developed, some of which — as for Apéry's constant — are unexpectedly fast.

Some constants differ so much from the usual kind that a new notation has been invented to represent them reasonably. Graham's number illustrates this as Knuth's up-arrow notation is used.

It may be of interest to represent them using continued fractions to perform various studies, including statistical analysis. Many mathematical constants have an analytic form, that is they can be constructed using well-known operations that lend themselves readily to calculation. Not all constants have known analytic forms, though; Grossman's constant and Foias' constant are examples.

Symbolizing and naming of constants

Symbolizing constants with letters is a frequent means of making the notation more concise. A common convention, instigated by René Descartes in the 17th century and Leonhard Euler in the 18th century, is to use lower case letters from the beginning of the Latin alphabet ${\displaystyle a,b,c,\dots }$ or the Greek alphabet ${\displaystyle \alpha ,\beta ,\gamma ,\dots }$ when dealing with constants in general.

However, for more important constants, the symbols may be more complex and have an extra letter, an asterisk, a number, a lemniscate or use different alphabets such as Hebrew, Cyrillic or Gothic.

Sometimes, the symbol representing a constant is a whole word. For example, American mathematician Edward Kasner's 9-year-old nephew coined the names googol and googolplex.

${\displaystyle \mathrm{g}\mathrm{o}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{l}={10}^{100},\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{l}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{x}={10}^{\mathrm{g}\mathrm{o}\mathrm{o}\mathrm{g}\mathrm{o}\mathrm{l}}={10}^{{10}^{100}}}$

Other names are either related to the meaning of the constant (universal parabolic constant, twin prime constant, ...) or to a specific person (Sierpiński's constant, Josephson constant, and so on).

Selected mathematical constants

Abbreviations used:

Gen – General, NuT – Number theory, ChT – Chaos theory, Com – Combinatorics, Ana – Mathematical analysis