Euler function

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

\phi(q)=\prod_{k=1}^\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.

[edit] Properties

The coefficient p(k) in the Maclaurin series for 1 / φ(q) gives the number of all partitions of k. That is,

\frac{1}{\phi(q)}=\sum_{k=0}^\infty p(k) q^k

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

The Euler identity is

\phi(q)=\sum_{n=-\infty}^\infty (-1)^n q^{(3n^2-n)/2}

Note that (3n2 − n) / 2 is a pentagon number.

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

φ(q) = q − 1 / 24η(τ)

where q = e2πiτ is the square of the nome.

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

[edit] References