Weber modular function
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In mathematics, the Weber modular functions are a family of three modular functions f, f1, and f2, studied by Heinrich Martin Weber.
Contents
Definition
Let where τ is an element of the upper half-plane.
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} \mathfrak{f}(\tau) &= q^{-\frac{1}{48}}\prod_{n>0}(1+q^{n-\frac{1}{2}}) = e^{-\frac{\pi\rm{i}}{24}}\frac{\eta\big(\frac{\tau+1}{2}\big)}{\eta(\tau)}=\frac{\eta^2(\tau)}{\eta\big(\tfrac{\tau}{2}\big)\eta(2\tau)}\\ \mathfrak{f}_1(\tau) &= q^{-\frac{1}{48}}\prod_{n>0}(1-q^{n-\frac{1}{2}}) = \frac{\eta\big(\tfrac{\tau}{2}\big)}{\eta(\tau)}\\ \mathfrak{f}_2(\tau) &= \sqrt2\, q^{-\frac{1}{24}}\prod_{n>0}(1+q^{n})= \frac{\sqrt2\,\eta(2\tau)}{\eta(\tau)} \end{align}
where is the Dedekind eta function. Note the eta quotients immediately imply that,
The transformation τ → –1/τ fixes f and exchanges f1 and f2. So the 3-dimensional complex vector space with basis f, f1 and f2 is acted on by the group SL2(Z).
Relation to theta functions
Let the argument of the Jacobi theta function be the nome . Then,
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} \mathfrak{f}(\tau) &= \sqrt{\frac{\theta_3(0,q)}{\eta(\tau)}} \\ \mathfrak{f}_1(\tau) &= \sqrt{\frac{\theta_4(0,q)}{\eta(\tau)}} \\ \mathfrak{f}_2(\tau) &= \sqrt{\frac{\theta_2(0,q)}{\eta(\tau)}} \\ \end{align}
Thus,
which is simply a consequence of the well known identity,
Relation to j-function
The three roots of the cubic equation,
where j(τ) is the j-function are given by . Also, since,
then,
See also
References
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