Mixing
Functions whose input is the same as the output?
$begingroup$
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 Jan 3 at 9:55
Tito Piezas IIITito Piezas III
26.9k365169
$endgroup$
add a comment |
$begingroup$
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 Jan 3 at 9:55
Tito Piezas IIITito Piezas III
26.9k365169
$endgroup$
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
add a comment |
$begingroup$
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 Jan 3 at 9:55
Tito Piezas IIITito Piezas III
26.9k365169
$endgroup$
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 Jan 3 at 9:55
Tito Piezas IIITito Piezas III
26.9k365169
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
26.9k365169
edited Jan 4 at 12:05
Tito Piezas III
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
26.9k365169
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
26.9k365169
asked Jan 3 at 9:55
Tito Piezas IIITito Piezas III
26.9k365169
26.9k365169
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
add a comment |
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10
add a comment |
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$begingroup$
Essentially the same eta quotients are used in this post.
$endgroup$
– Tito Piezas III
Jan 4 at 12:10