# Functions whose input is the same as the output?

Given the Dedekind eta function $$\eta(\tau)$$ and complex number $$\tau$$. I came across these family of functions,

$${f_2(\tau)= \frac{i}{\sqrt{2}}\frac{\,_2F_1\left(\tfrac14,\tfrac34,1,\,1-\alpha_2\right)}{\,_2F_1\left(\tfrac14,\tfrac34,1,\,\alpha_2\right)}=\tau}$$

$${f_3(\tau)= \frac{i}{\sqrt{3}}\frac{\,_2F_1\left(\tfrac13,\tfrac23,1,\,1-\alpha_3\right)}{\,_2F_1\left(\tfrac13,\tfrac23,1,\,\alpha_3\right)}=\tau}$$

$${f_4(\tau)= \frac{i}{\sqrt{4}}\frac{\,_2F_1\left(\tfrac12,\tfrac12,1,\,1-\alpha_4\right)}{\,_2F_1\left(\tfrac12,\tfrac12,1,\,\alpha_4\right)}=\tau}$$

where,

$$\alpha_2 =\frac{64}{64+\Big(\frac{\eta(\tau)}{\eta(2\tau)}\Big)^{24}},\quad \alpha_3 =\frac{27}{27+\Big(\frac{\eta(\tau)}{\eta(3\tau)}\Big)^{12}},\quad \alpha_4 =\frac{16}{16+\Big(\frac{\eta(\tau)}{\eta(4\tau)}\Big)^{8}},$$

So the input variable is $$\tau$$ and the output is also $$\tau$$. Presumably these are identity functions $$f(x)=x$$?

Q: What are other not-so-trivial examples of identity functions?

P.S. There is a $$f_1(\tau)$$ using $$\,_2F_1\left(\tfrac16,\tfrac56,1,\,\alpha_1\right)$$ but it uses the j-function, instead of the Dedekind eta function.

• Essentially the same eta quotients are used in this post. – Tito Piezas III Jan 4 at 12:10