Modulus of phi on the complex plane, colored so that black=0, red=4

For other meanings, see Euler function (disambiguation).

In mathematics, the Euler function is given by Eulers Q-series on the complex plane (600x600 pixels) Detailed description This picture shows the modulus on the complex plane, inside the unit circle , where is Eulers q-series for the Partition of an integer. ... Eulers Q-series on the complex plane (600x600 pixels) Detailed description This picture shows the modulus on the complex plane, inside the unit circle , where is Eulers q-series for the Partition of an integer. ... In mathematics, Euler function may refer to several unrelated or loosely related functions, all named after Leonhard Euler. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...

Named after Leonhard Euler, it is a prototypical example of a q-series, a modular form, and provides the prototypical example of a relation between combinatorics and complex analysis. Leonhard Euler by Emanuel Handmann Leonhard Euler [oilər] (April 15, 1707–September 18, 1783) was a Swiss mathematician and physicist. ... In mathematics, a q-series, also sometimes called a q-shifted factorial, is defined as It is usually considered first as a formal power series; it is also an analytic function of q, in the unit disc. ... Modular form - Wikipedia /**/ @import /skins-1. ... Combinatorics is a odd branch of mathematics that studies collections (usually finite) then constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), with finding largest, smallest, or optimal objects (extremal combinatorics and combinatorial optimization), and with finding algebraic structures these objects may have (algebraic combinatorics). ... This article may be too technical for most readers to understand. ...

Properties

The coefficient of q^k in the Maclaurin series for 1 / φ(q) gives the number of all partitions of k. That is, As the degree of the taylor series rises, it approaches the correct function. ... In mathematics, a partition of a positive integer n is a way of writing n as a sum of positive integers. ...

where p(k) is the partition function of k. In number theory, the partition function p(n) represents the number of possible partitions of a natural number n, which is to say the number of distinct (and order independent) ways of representing n as a sum of natural numbers. ...

The Euler identity is In mathematics, the pentagonal number theorem, originally due to Euler, states that This theorem can be given a combinatorial interpretation in terms of partitions. ...

Note that (3n² − n) / 2 is a pentagon number. In mathematics, a polygonal number is a number that can be arranged as a regular polygon. ...

The Euler function is related to the Dedekind eta function through a Ramanujan identity as The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ... Ramanujan Srinivasa Aiyangar Ramanujan (Tamil: ஸ்ரீனிவாஸ ஐயங்கார் ராமானுஜன்) (December 22, 1887 – April 26, 1920) was a groundbreaking Indian mathematician. ...

φ(q) = q ^{− 1 / 24}η(τ)

where q = e^2πiτ is the square of the nome. In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ...

Note that both functions have the symmetry of the modular group. In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...

References

• Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. ISBN 0-387-90163-9