Evection

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

For other meanings, see List of topics named after Leonhard Euler.

In mathematics, the Euler function is given by

$\varphi (q)=\prod _{k=1}^{\mathrm{\infty }}(1-q^k).$

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.

Properties

The coefficient $p(k)$ in the formal power series expansion for $1/\varphi (q)$ gives the number of all partitions of k. That is,

$\frac{1}{\varphi (q)}=\sum _{k=0}^{\mathrm{\infty }}p(k)q^k$

where $p(k)$ is the partition function of k.

The Euler identity, also known as the Pentagonal number theorem is

$\varphi (q)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(-1{)}^n{q}^{(3n^2-n)/2}.$

Note that $(3n^2-n)/2$ is a pentagonal number.

The Euler function is related to the Dedekind eta function through a Ramanujan identity as

$\varphi (q)=q^{-\frac{1}{24}}\eta (\tau )$

where $q=e^{2\pi i\tau }$ is the square of the nome.

Note that both functions have the symmetry of the modular group.

The Euler function may be expressed as a Q-Pochhammer symbol:

$\varphi (q)=(q;q{)}_{\mathrm{\infty }}$

The logarithm of the Euler function is the sum of the logarithms in the product expression, each of which may be expanded about q=0, yielding:

$\ln (\varphi (q))=-\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}\frac{q^n}{1-q^n}$

which is a Lambert series with coefficients -1/n. The logarithm of the Euler function may therefore be expressed as:

$\ln (\varphi (q))=\sum _{n=1}^{\mathrm{\infty }}{b}_n{q}^n$

where

$b_n=-\sum _{d|n}\frac{1}{d}=$ -[1/1, 3/2, 4/3, 7/4, 6/5, 12/6, 8/7, 15/8, 13/9, 18/10, ...] (see OEIS A000203)

On account of the following identity,

$\sum _{d|n}d=\sum _{d|n}\frac{n}{d}$

this may also be written as

$\ln (\varphi (q))=-\sum _{n=1}^{\mathrm{\infty }}\frac{q^n}{n}\sum _{d|n}d$

Special values

The next identities come from Ramanujan's lost notebook, Part V, p. 326.

$\varphi (e^{-\pi })=\frac{e^{\pi /24}\Gamma \left(\frac{1}{4}\right)}{2^{7/8}{\pi }^{3/4}}$

$\varphi (e^{-2\pi })=\frac{e^{\pi /12}\Gamma \left(\frac{1}{4}\right)}{2{\pi }^{3/4}}$

$\varphi (e^{-4\pi })=\frac{e^{\pi /6}\Gamma \left(\frac{1}{4}\right)}{2^{11/8}{\pi }^{3/4}}$

$\varphi (e^{-8\pi })=\frac{e^{\pi /3}\Gamma \left(\frac{1}{4}\right)}{2^{29/16}{\pi }^{3/4}}(\sqrt{2}-1{)}^{1/4}$

References

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