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,
$${f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-alpha_2right)}{,_2F_1left(tfrac14,tfrac34,1,,alpha_2right)}=tau}$$
$${f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-alpha_3right)}{,_2F_1left(tfrac13,tfrac23,1,,alpha_3right)}=tau}$$
$${f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-alpha_4right)}{,_2F_1left(tfrac12,tfrac12,1,,alpha_4right)}=tau}$$
where,
$$alpha_2 =frac{64}{64+Big(frac{eta(tau)}{eta(2tau)}Big)^{24}},quad
alpha_3 =frac{27}{27+Big(frac{eta(tau)}{eta(3tau)}Big)^{12}},quad
alpha_4 =frac{16}{16+Big(frac{eta(tau)}{eta(4tau)}Big)^{8}},$$
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 $,_2F_1left(tfrac16,tfrac56,1,,alpha_1right)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked yesterday
Tito Piezas III
26.8k365169
add a comment |
Given the Dedekind eta function $eta(tau)$ and complex number $tau$. I came across these family of functions,
$${f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-alpha_2right)}{,_2F_1left(tfrac14,tfrac34,1,,alpha_2right)}=tau}$$
$${f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-alpha_3right)}{,_2F_1left(tfrac13,tfrac23,1,,alpha_3right)}=tau}$$
$${f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-alpha_4right)}{,_2F_1left(tfrac12,tfrac12,1,,alpha_4right)}=tau}$$
where,
$$alpha_2 =frac{64}{64+Big(frac{eta(tau)}{eta(2tau)}Big)^{24}},quad
alpha_3 =frac{27}{27+Big(frac{eta(tau)}{eta(3tau)}Big)^{12}},quad
alpha_4 =frac{16}{16+Big(frac{eta(tau)}{eta(4tau)}Big)^{8}},$$
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 $,_2F_1left(tfrac16,tfrac56,1,,alpha_1right)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked yesterday
Tito Piezas III
26.8k365169
Essentially the same eta quotients are used in this post.
– Tito Piezas III
16 hours ago
add a comment |
Given the Dedekind eta function $eta(tau)$ and complex number $tau$. I came across these family of functions,
$${f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-alpha_2right)}{,_2F_1left(tfrac14,tfrac34,1,,alpha_2right)}=tau}$$
$${f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-alpha_3right)}{,_2F_1left(tfrac13,tfrac23,1,,alpha_3right)}=tau}$$
$${f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-alpha_4right)}{,_2F_1left(tfrac12,tfrac12,1,,alpha_4right)}=tau}$$
where,
$$alpha_2 =frac{64}{64+Big(frac{eta(tau)}{eta(2tau)}Big)^{24}},quad
alpha_3 =frac{27}{27+Big(frac{eta(tau)}{eta(3tau)}Big)^{12}},quad
alpha_4 =frac{16}{16+Big(frac{eta(tau)}{eta(4tau)}Big)^{8}},$$
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 $,_2F_1left(tfrac16,tfrac56,1,,alpha_1right)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked yesterday
Tito Piezas III
26.8k365169
Given the Dedekind eta function $eta(tau)$ and complex number $tau$. I came across these family of functions,
$${f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-alpha_2right)}{,_2F_1left(tfrac14,tfrac34,1,,alpha_2right)}=tau}$$
$${f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-alpha_3right)}{,_2F_1left(tfrac13,tfrac23,1,,alpha_3right)}=tau}$$
$${f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-alpha_4right)}{,_2F_1left(tfrac12,tfrac12,1,,alpha_4right)}=tau}$$
where,
$$alpha_2 =frac{64}{64+Big(frac{eta(tau)}{eta(2tau)}Big)^{24}},quad
alpha_3 =frac{27}{27+Big(frac{eta(tau)}{eta(3tau)}Big)^{12}},quad
alpha_4 =frac{16}{16+Big(frac{eta(tau)}{eta(4tau)}Big)^{8}},$$
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 $,_2F_1left(tfrac16,tfrac56,1,,alpha_1right)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
complex-analysis functions terminology special-functions hypergeometric-function
asked yesterday
Tito Piezas III
26.8k365169
asked yesterday
Tito Piezas III
26.8k365169
edited 16 hours ago
asked yesterday
Tito Piezas III
26.8k365169
asked yesterday
Tito Piezas III
26.8k365169
asked yesterday
Tito Piezas III
26.8k365169
26.8k365169
Essentially the same eta quotients are used in this post.
– Tito Piezas III
16 hours ago
add a comment |
Essentially the same eta quotients are used in this post.
– Tito Piezas III
16 hours ago
Essentially the same eta quotients are used in this post.
– Tito Piezas III
16 hours ago
Essentially the same eta quotients are used in this post.
– Tito Piezas III
16 hours ago
add a comment |
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Essentially the same eta quotients are used in this post.
– Tito Piezas III
16 hours ago