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,
$$large{f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-frac{64}{64+x_2}right)}{,_2F_1left(tfrac14,tfrac34,1,,frac{64}{64+x_2}right)}=tau}$$
$$large{f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-frac{27}{27+x_3}right)}{,_2F_1left(tfrac13,tfrac23,1,,frac{27}{27+x_3}right)}=tau}$$
$$large{f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-frac{16}{16+x_4}right)}{,_2F_1left(tfrac12,tfrac12,1,,frac{16}{16+x_4}right)}=tau}$$
where,
$$x_2 =Big(frac{eta(tau)}{eta(2tau)}Big)^{24},quad
x_3 =Big(frac{eta(tau)}{eta(3tau)}Big)^{12},quad
x_4 =Big(frac{eta(tau)}{eta(4tau)}Big)^{8}$$
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 $,_2F_1left(tfrac16,tfrac56,1,,alpharight)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked 19 hours ago
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,
$$large{f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-frac{64}{64+x_2}right)}{,_2F_1left(tfrac14,tfrac34,1,,frac{64}{64+x_2}right)}=tau}$$
$$large{f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-frac{27}{27+x_3}right)}{,_2F_1left(tfrac13,tfrac23,1,,frac{27}{27+x_3}right)}=tau}$$
$$large{f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-frac{16}{16+x_4}right)}{,_2F_1left(tfrac12,tfrac12,1,,frac{16}{16+x_4}right)}=tau}$$
where,
$$x_2 =Big(frac{eta(tau)}{eta(2tau)}Big)^{24},quad
x_3 =Big(frac{eta(tau)}{eta(3tau)}Big)^{12},quad
x_4 =Big(frac{eta(tau)}{eta(4tau)}Big)^{8}$$
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 $,_2F_1left(tfrac16,tfrac56,1,,alpharight)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked 19 hours ago
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,
$$large{f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-frac{64}{64+x_2}right)}{,_2F_1left(tfrac14,tfrac34,1,,frac{64}{64+x_2}right)}=tau}$$
$$large{f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-frac{27}{27+x_3}right)}{,_2F_1left(tfrac13,tfrac23,1,,frac{27}{27+x_3}right)}=tau}$$
$$large{f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-frac{16}{16+x_4}right)}{,_2F_1left(tfrac12,tfrac12,1,,frac{16}{16+x_4}right)}=tau}$$
where,
$$x_2 =Big(frac{eta(tau)}{eta(2tau)}Big)^{24},quad
x_3 =Big(frac{eta(tau)}{eta(3tau)}Big)^{12},quad
x_4 =Big(frac{eta(tau)}{eta(4tau)}Big)^{8}$$
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 $,_2F_1left(tfrac16,tfrac56,1,,alpharight)$ but it uses the j-function, instead of the Dedekind eta function.
complex-analysis functions terminology special-functions hypergeometric-function
asked 19 hours ago
Tito Piezas III
26.8k365169
Given the Dedekind eta function $eta(tau)$ and complex number $tau$. I came across these family of functions,
$$large{f_2(tau)= frac{i}{sqrt{2}}frac{,_2F_1left(tfrac14,tfrac34,1,,1-frac{64}{64+x_2}right)}{,_2F_1left(tfrac14,tfrac34,1,,frac{64}{64+x_2}right)}=tau}$$
$$large{f_3(tau)= frac{i}{sqrt{3}}frac{,_2F_1left(tfrac13,tfrac23,1,,1-frac{27}{27+x_3}right)}{,_2F_1left(tfrac13,tfrac23,1,,frac{27}{27+x_3}right)}=tau}$$
$$large{f_4(tau)= frac{i}{sqrt{4}}frac{,_2F_1left(tfrac12,tfrac12,1,,1-frac{16}{16+x_4}right)}{,_2F_1left(tfrac12,tfrac12,1,,frac{16}{16+x_4}right)}=tau}$$
where,
$$x_2 =Big(frac{eta(tau)}{eta(2tau)}Big)^{24},quad
x_3 =Big(frac{eta(tau)}{eta(3tau)}Big)^{12},quad
x_4 =Big(frac{eta(tau)}{eta(4tau)}Big)^{8}$$
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 $,_2F_1left(tfrac16,tfrac56,1,,alpharight)$ 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 19 hours ago
Tito Piezas III
26.8k365169
asked 19 hours ago
Tito Piezas III
26.8k365169
edited 17 hours ago
asked 19 hours ago
Tito Piezas III
26.8k365169
asked 19 hours ago
Tito Piezas III
26.8k365169
asked 19 hours ago
Tito Piezas III
26.8k365169
26.8k365169
add a comment |
add a comment |
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