User:Singular Modulus

The complete elliptic integral of the first kind is defined as
 * $$K(x) = \int_0^{\pi/2} \frac{d \theta}{\sqrt{1 - x^2 \sin^2 \theta}}=\frac{\pi}{2}{}_2F_1\left(\frac{1}{2},\frac{1}{2};1;x^2\right)$$

Using this function the Singular Modulus kr with r>0 is defined as the solution of the equation


 * $$ \frac{K(k'_r)}{K(k_r)}=\sqrt{r} $$


 * $$ k'_r=\sqrt{1-k_r^2} $$

The argument k'r is called complementary modulus. In the case when r is positive rational the elliptic singular modulus kr is algebraic number. For example


 * $$ k'_1=\frac{1}{\sqrt{2}} $$
 * $$k'_2=\sqrt{2(\sqrt{2}-1)}$$
 * $$k'_3=\sqrt{\frac{2+\sqrt{3}}{4}}$$
 * $$k'_4=2^{5/4}(\sqrt{2}-1)$$
 * $$k'_5=\sqrt{\frac{1+2\sqrt{\sqrt{5}-2}}{2}}$$
 * $$k_6=(\sqrt{3}-\sqrt{2})(2-\sqrt{3})$$
 * $$k_7=\sqrt{\frac{8-3\sqrt{7}}{16}}$$
 * $$k_9=\frac{1}{\sqrt{2}}\sqrt{(2-\sqrt{3})(\sqrt{2}-\sqrt[4]{3})^2}$$
 * $$k_{10}=(\sqrt{10}-3)(\sqrt{2}-1)^2$$
 * $$k_{12}=(\sqrt{3}-\sqrt{2})^2(\sqrt{2}-1)^2$$
 * $$k_{15}=\frac{1}{8\sqrt{2}}(2-\sqrt{3})(\sqrt{5}-\sqrt{3})(3-\sqrt{5})$$
 * $$k_{16}=\frac{(\sqrt{2}-1)^2}{(\sqrt[4]{2}+1)^4}$$
 * $$k_{18}=(2-\sqrt{3})^2(\sqrt{2}-1)^3$$
 * $$k_{25}=\frac{1}{\sqrt{2}}(\sqrt{5}-2)(3-2\sqrt[4]{5})$$

These are some of the first values in positive integers. The calculation of these and other houndred of values are made by various scientists such as Weber and Ramanujan. These values have been verifyed and reproven by Berndt the Borwein brothers and other scientists in nowdays because of their use in many theories. An example of such evaluation is
 * $$G_{765}=\left(\sqrt{\frac{96+11\sqrt{79}}{4}}+\sqrt{\frac{100+11\sqrt{79}}{4}}\right)^{1/2}\left(\sqrt{\frac{141+16\sqrt{79}}{2}}+\sqrt{\frac{143+16\sqrt{79}}{2}}\right)^{1/2}$$

In general the Weber invariants Gr and gr are defined when q=e-π sqrt(r) by
 * $$G_r=2^{-1/4}q^{-1/24}\prod^{\infty}_{n=1}(1+q^{2n-1})$$
 * $$g_r=2^{-1/4}q^{-1/24}\prod^{\infty}_{n=1}(1-q^{2n-1})$$

and related with the singular modulus by
 * $$G_r=\left(2k_rk'_r\right)^{-1/12}$$
 * $$g_r=\left(2k_r(k'_r)^{-2}\right)^{-1/12}$$

David Broadhurst give evaluations of kr in r=3(mod8) and by his method evaluate k2317723.

Modular Equations and Singular Modulus
Different degree of singular modulus kn^2r and kr are related with equations which they called modular equations. The most simple modular equation is that of degree 2
 * $$k_{4r}=\frac{1-k'_r}{1+k'_r}$$

The 3rd degree modular equation is
 * $$\sqrt{k_rk_{9r}}+\sqrt{k'_rk'_{9r}}=1$$

The 5th degree modular equation is
 * $$k_rk_{25r}+k'_rk'_{25r}+2\sqrt[3]{4(k_rk_{25r}k'_rk'_{25r}}=1$$

The first two of these equations are solvable in radicals. The 5th degree modular equation is not solvable, but can reduce in the form
 * $$k_{25^nr_0}=\Phi_n(k_{r_0},k_{r_0/25}), n\in Z$$

which can be described as
 * $$Q(x)=\frac{\left(-1-e^{\frac{1}{5}y}+e^{\frac{2}{5} y}\right)^5}{\left(e^{\frac{1}{5}y}-e^{\frac{2}{5}y}+2 e^{\frac{3}{5} y}-3 e^{\frac{4}{5}y}+5 e^{y}+3 e^{\frac{6}{5}y}+2 e^{\frac{7}{5}y}+e^{\frac{8}{5} y}+e^{\frac{9}{5}y}\right)}$$

where
 * $$y=\textrm{arcsinh}\left(\frac{11+x}{2}\right)$$
 * $$U(x)=\sqrt{-\frac{5}{3 x^2}+\frac{25}{3 x^2 h(x)}+\frac{x^4}{h(x)}+\frac{h(x)}{3 x^2}}$$

where
 * $$h(x)=\sqrt[3]{-125-9x^6+3\sqrt{3}\sqrt{-125x^6-22x^{12}-x^{18}}}$$
 * $$U^{*}(x)=\sqrt{-\frac{1}{2 x^2}+\frac{x^4}{2}+\frac{\sqrt{1+18 x^6+x^{12}}}{2 x^2}}$$
 * $$P(x)=U(Q^{1/6}(U^{*6}(x)))$$
 * $$k_{25^nr_0}=\sqrt{1/2-1/2\sqrt{1-4\left(k_{r_0}k'_{r_0}\right)^2\prod^{n}_{j=1}P^{(j)}\left[\sqrt[12]{\frac{k_{r_0}k'_{r_0}}{k_{r_0/25}k'_{r_0/25}}}\right]^{24}}}$$

The reducing problem of finding k25r, is to solve the depressed equation after named by Hermite
 * $$u^6-\nu^6+5u^2\nu^2(u^2-\nu^2)+4u\nu(1-u^4\nu^4)=0$$

where u=(kr)1/4 and v=(k25r)1/4. The function kr is also connected to theta functions by the relation
 * $$k_r=\frac{\theta^2_2(q)}{\theta^2_3(q)}$$

where
 * $$\theta_2(q)=\sum^{\infty}_{n=-\infty}q^{(n+\frac{1}{2})^2}$$

and
 * $$\theta_3(q)=\sum^{\infty}_{n=-\infty}q^{n^2}$$

The 7th degree modular equation is
 * $$\sqrt[4]{k_rk_{49r}}+\sqrt[4]{k'_rk'_{49r}}=1$$

This form of 7th degree modular equation is due to Guetzlaff in 1834. Fiedler in 1835 and Schroter in 1854 also proved this modular equation. More complicated modular equations of degree 7 have been discovered by Schlafli, Klein, Sohncke and Russell.

The 11th degree modular equation is
 * $$\sqrt{k_rk_{121r}}+\sqrt{k'_rk'_{121r}}+2\sqrt[6]{k_rk_{121r}k'_rk'_{121r}}=1$$

The 23rd degree modular equation is
 * $$\sqrt[4]{k_rk_{529r}}+\sqrt[4]{k'_rk'_{529r}}+\sqrt[12]{256(k_rk_{529r}k'_rk'_{529r})}=1$$

Elliptic Integrals at Singular Values
The class number is given by
 * $$h(d)=-\frac{w(d)}{2|d|}\sum^{|d|-1}_{l=1}\left(\frac{d}{l}\right)l$$

and w(d)=(6 if d=-3), (4 if d=-4), (2 else). When N is natural number>3, N=3(mod4) and square free with class number h(-N)=1 then there 5 such cases and for these cases hold
 * $$K(k_N)=\frac{G_N^2}{2}\sqrt{\frac{\pi}{N}\prod^{N}_{n=1}\Gamma\left(\frac{n}{N}\right)^{(\frac{-N}{n})}}$$

In general for natural N=3(mod4) and square free, then λN is an algebraic number in which the minimal polynomial can reduced into a polynomial of degree h(-N) and holds
 * $$K(k_N)=\frac{G_N^2}{2}\sqrt{\frac{\pi}{N}\left(\lambda_N^4\prod^{N}_{n=1}\Gamma\left(\frac{n}{N}\right)^\left(\frac{-N}{n}\right)\right)^{1/h(-N)}}$$

In the case of h(-N)=0 then λN satisfies a-bxm=0, with a,b,m positive integers. Note also that in this case there are infinite denumerable N and holds
 * $$\prod^{N}_{n=1}\Gamma\left(\frac{n}{N}\right)^{\left(\frac{-N}{n}\right)}=\left(\frac{b}{a}\right)^{4/m}$$

Properties and Relations
The rth singular value satisfies
 * $$AGM(1,k'_r)=\sqrt{r}AGM(1,k_r)$$

where the arithemtic-geometric mean (AGM) is obtained by iterating the rapidly convergence process
 * $$AGM\left(a,b\right)=AGM\left((a+b)/2,\sqrt{ab}\right)$$
 * $$k_r=4q^{1/2}exp\left(-4\sum^{\infty}_{n=1}\left(\sum_{d|n}\frac{(-1)^{d+n/d}}{d}\right)q^n\right)$$

Also the integral of the Dedekind-eta function η(τ) and the Rogers-Ramanujan continued fraction R(q), q=e-πsqrt(r) are related with the singular modulus by
 * $$\pi\int^{+\infty}_{\sqrt{r}}\eta(it/2)^4dt=5\int^{R(q)}_{0}\frac{dx}{x\sqrt[6]{x^{-5}-11-x^5}}=3\sqrt[3]{2k_r}\cdot{}_2F_1\left[\frac{1}{3},\frac{1}{6};\frac{7}{6};k_r^2\right]$$

Also if wr=sqrt(krk25r), w'r=sqrt(k'rk'25r) , then
 * $$A_r=R^{-5}(q^2)-11-R^5(q^2)=\frac{(k_rk'_r)^2}{(w_rw'_r)^2}\left(\frac{w_r}{k_r}+\frac{w'_r}{k'_r}-\frac{w_rw'_r}{k_rk'_r}\right)^3$$

The Ramanujan's cubic continued fraction is
 * $$V(q)=\frac{q^{1/3}}{1+}\frac{q+q^2}{1+}\frac{q^2+q^4}{1+}\frac{q^3+q^6}{1+}\ldots$$
 * $$T=\sqrt{1-8V^3(q)}$$
 * $$(k_r)^2=\frac{(1-T)(3+T)^3}{(1+T)(3-T)^3}$$

The Ramanujan-Gollnitz-Gordon is
 * $$H(q)=\frac{q^{1/2}}{(1+q)+}\frac{q^2}{(1+q^3)+}\frac{q^4}{(1+q^5)+}\ldots$$

For this hold
 * $$H(q)=\sqrt{t^2+1}-t$$

and
 * $$t=\frac{k_r}{1-k'_r}$$

also if kr=k
 * $$\frac{dH(q)}{dk}=\frac{\sqrt{1-k'}}{k'(k\sqrt{2}+2\sqrt{1-k'})}$$
 * $$\frac{dq}{dk}=\frac{-q\pi^2}{2k(1-k^2)K(k)^2}$$

If m is integer, then some theta functions can evaluated explicity
 * $$\sum^{\infty}_{n=-\infty}q^{n^2+2mn}=2^{1/2}q^{-m^2}\sqrt{\frac{K(k_r)}{\pi}}$$

also for k11=k, k12=sqrt(1-k112), k21=(2-k112-2k12)/k112, k22=sqrt(1-k212) then
 * $$\sum^{\infty}_{n=-\infty}q^{n^2+(2m+1)n}=2^{5/6}q^{-(2m+1)^2/4}\frac{(k_{11}k_{12}k_{21})^{1/6}}{(k_{22})^{1/3}}\sqrt{\frac{K(k_r)}{\pi}}$$

Also if q=e-πsqrt(r), the following evaluation of Dedekind eta function f is valid
 * $$f(-q)=\prod^{\infty}_{n=1}(1-q^n)=2^{1/3}\pi^{-1/2}q^{-1/24}k^{1/12}k'^{1/3}K(k)^{1/2}$$

A root of the equation
 * $$\frac{b^2}{20a}+bX+aX^2=cX^{5/3}$$

is
 * $$X=\frac{b}{250a}\left(R(e^{-2\pi\sqrt{r}})^{-5}-11-R(e^{-2\pi\sqrt{r}})^5\right)=\frac{b}{250a}A_r$$

where the j-class invariant is jr=250c3a-2b-1, in order to generate the Roger's-Ramanujan continued fraction.

A new elliptic integral that rise from Rogers-Ramanujan continued fraction is
 * $$\int^{y}_{0}\frac{dt}{t\sqrt[6]{t^{-5}-11-t^5}}=A$$

then
 * $$y=R\left(e^{-\pi\sqrt{m(A)}}\right)$$

where m(x) function is defined as
 * $$x=\pi\int^{+\infty}_{\sqrt{m(x)}}\eta(it/2)^4dt$$

and have the property
 * $$m(x)=r\Leftrightarrow x=3\sqrt[3]{2k_r}\cdot{}_2F_{1}\left[\frac{1}{3},\frac{1}{6};\frac{7}{6};k^2_r\right]$$

Another very interesting property of the Roger's-Ramanujan continued fraction is that transforms generalized integrals
 * $$\int^{b}_{a}f(-q)^4q^{-5/6}G(R(q))dq=5\int^{R(b)}_{R(a)}\frac{G(x)}{x\sqrt[6]{x^{-5}-11-x^5}}dx$$

In a similar way if we define mG by
 * $$x=\pi\int^{+\infty}_{\sqrt{m_G(x)}}\eta(it/2)^4G\left(R\left(e^{-\pi t}\right)\right)dt$$

then if y is such that
 * $$5\int^{y}_{0}\frac{G(t)dt}{t\sqrt[6]{t^{-5}-11-t^5}}=A$$

then
 * $$y=R\left(e^{-\pi\sqrt{m_G(A)}}\right)$$

and
 * $$y'(A)=\frac{R'(q_m)q_m}{\eta\left(\frac{i\sqrt{m_G(A)}}{2}\right)^4G(R(q_m))}$$

where
 * $$q_m=e^{-\pi\sqrt{m_G(A)}}$$

Other Modular Equations

 * $$u_{n,\nu}=\frac{g_{n^2\nu^2r}}{2 (g_{\nu^2r})^n}$$

For n=2
For n=2 u=u2,1 and v=u2,2 satisfy the modular equation
 * $$1+u^4-2u^2v^4=0$$

For u=u2,1 and v=u2,3 satisfy the modular equation
 * $$u^4+2 u v-2 u^3 v^3-v^4=0$$

For u=u2,1 and v=u2,4 satisfy the modular equation
 * $$-1-4 u^2-6 u^4-4 u^6-u^8+8 u^2 v^8+8 u^6 v^8=0$$

For u=u2,1 and v=u2,5 satisfy the modular equation
 * $$u^6+4 u v+5 u^4 v^2-5 u^2 v^4-4 u^5 v^5-v^6=0$$

u=u2,1 and v=u2,7 satisfy the modular equation
 * $$u^8-8 u v+28 u^2 v^2-56 u^3 v^3+70 u^4 v^4-56 u^5 v^5+28 u^6 v^6-8 u^7 v^7+v^8=0$$

The functions u=u2,1 and v=u2,11 satisfy the modular equation
 * $$u^{12}+32 u v-22 u^9 v+44 u^6 v^2+88 u^3 v^3+22 u^{11} v^3+165 u^8 v^4+132 u^5 v^5-44 u^2 v^6+44 u^{10} v^6$$
 * $$-132 u^7 v^7-165 u^4 v^8-22 u v^9-88 u^9 v^9-44 u^6 v^{10}+22 u^3 v^{11}-32 u^{11} v^{11}-v^{12}=0$$

For n=3
The functions u=u3,1 and v=u3,2 satisfy the modular equation
 * $$u^2+v-2 u v^2=0$$

The functions u=u3,1 and v=u3,3 satisfy the modular equation
 * $$u + u^2 + u^3 - v^3 + 2 u v^3 - 4 u^2 v^3=0$$

The functions u=u3,1 and v=u3,4 satisfy the modular equation
 * $$u^4+v+4 u^3 v+6 u^2 v^2-4 u v^3-8 u^3 v^4=0$$

The functions u=u3,1 and v=u3,5 satisfy the modular equation
 * $$u^6-u v-5 u^4 v+5 u^2 v^2+10 u^5 v^2-20 u^3 v^3-5 u v^4+20 u^4 v^4+10 u^2 v^5-16 u^5 v^5+v^6=0$$

The functions u=u3,1 and v=u3,6 satisfy the modular equation
 * $$u^2+2 u^3+3 u^4+2 u^5+u^6+v^3+2 u v^3+6 u^2 v^3+2 u^3 v^3+28 u^4 v^3+24 u^5 v^3-8 u v^6

+8 u^2 v^6-24 u^3 v^6-16 u^4 v^6-32 u^5 v^6=0$$ The functions u=u3,1 and v=u3,7 satisfy the modular equation
 * $$u^8-u v-7 u^4 v+28 u^6 v^2-56 u^5 v^3-7 u v^4+21 u^4 v^4+56 u^7 v^4-56 u^3 v^5+

+28 u^2 v^6+56 u^4 v^7-64 u^7 v^7+v^8=0$$ The functions u=u3,1 and v=u3,11 satisfy the modular equation
 * $$u^{12}-u v-11 u^4 v-22 u^7 v+22 u^{10} v-11 u^2 v^2-66 u^5 v^2+187 u^8 v^2-44 u^{11} v^2-22 u^3 v^3-154 u^6 v^3-44 u^9 v^3-11 uv^4-88 u^4 v^4$$
 * $$-154 u^7 v^4+748 u^{10} v^4-66 u^2 v^5-682 u^5v^5+308 u^8 v^5-352 u^{11} v^5-154 u^3 v^6-154 u^6 v^6+1232 u^9v^6-22 u v^7-154 u^4 v^7$$
 * $$-2728 u^7 v^7+2112 u^{10} v^7+187 u^2v^8+308 u^5 v^8-1408 u^8 v^8+1408 u^{11} v^8-44 u^3 v^9+1232 u^6v^9-1408 u^9 v^9+22 u v^{10}$$
 * $$+748 u^4 v^{10}+2112 u^7 v^{10}-2816u^{10} v^{10}-44 u^2 v^{11}-352 u^5 v^{11}+1408 u^8 v^{11}

-1024u^{11} v^{11}+v^{12}=0$$

For n=5
The functions u=u5,1 and v=u5,2 satisfy the modular equation
 * $$u^2+v+2 u v-4 u v^2=0$$

The functions u=u5,1 and v=u5,3 satisfy the modular equation
 * $$u^4-u v-3 u^2 v-3 u v^2-3 u^2 v^2+12 u^3 v^2+12 u^2 v^3-16 u^3 v^3+v^4=0$$

The functions u=u5,1 and v=u5,4 satisfy the modular equation
 * $$u^4+v+4 u v+2 u^2 v+4 u^3 v+8 u v^2+36 u^2 v^2-8 u^3 v^2-16 u v^3-32 u^2 v^3+64 u^3 v^3-64 u^3 v^4=0$$

The functions u=u5,1 and v=u5,5 satisfy the modular equation
 * $$u-4 u^2+6 u^3-4 u^4+u^5+5 u v-15 u^2 v+15 u^3 v-5 u^4 v+5 u v^2+10 u^2 v^2-35 u^3 v^2+20 u^4 v^2+5 u v^3+35 u^2 v^3$$
 * $$+40 u^3 v^3-80 u^4 v^3+5 u v^4+60 u^2 v^4

+240 u^3 v^4+320 u^4 v^4-v^5-16 u v^5-96 u^2 v^5-256 u^3 v^5-256 u^4 v^5=0$$ The functions u=u5,1 and v=u5,6 satisfy the modular equation
 * $$u^8+u^2 v+6 u^3 v+9 u^4 v+8 u^5 v+12 u^6 v-24 u^7 v+3 u^2 v^2+18 u^3 v^2-15 u^4 v^2-112 u^5 v^2+192 u^6 v^2-48 u^7 v^2$$
 * $$+48 u^3 v^3+12 u^4 v^3-672 u^5 v^3+448 u^6 v^3+128 u^7 v^3+v^4+8 u v^4+24 u^2 v^4+128 u^3 v^4+448 u^4 v^4-48 u^5 v^4$$
 * $$-240 u^6 v^4-576 u^7 v^4-48 u^2 v^5-448 u^3 v^5-512 u^4 v^5+768 u^5 v^5-1152 u^6 v^5+1536 u^7 v^5+48 u v^6+240 u^2 v^6$$
 * $$+192 u^3 v^6++384 u^4 v^6+768 u^6 v^6-1024 u^7 v^6-64 u v^7-192 u^2 v^7-512 u^4 v^7+256 u^4 v^8=0$$

For n=7
The functions u=u7,1 and v=u7,2 satisfy the modular equation
 * $$u^4-2 u^2 v-2 u^3 v+v^2-2 u v^2-12 u^2 v^2-16 u^3 v^2-16 u v^3-16 u^2 v^3+64 u^2 v^4=0$$

The functions u=u7,1 and v=u7,3 satisfy the modular equation
 * $$u^8-2 u^5 v-3 u^6 v-3 u^7 v+u^2 v^2-6 u^3 v^2-6 u^4 v^2-12 u^5 v^2+75 u^6 v^2-24 u^7 v^2-6 u^2 v^3+33 u^3 v^3$$
 * $$+6 u^4 v^3-12 u^5 v^3-96 u^6 v^3-128 u^7 v^3-6 u^2 v^4+6 u^3 v^4+994 u^4 v^4+48 u^5 v^4-384 u^6 v^4-2 u v^5$$
 * $$-12 u^2 v^5-12 u^3 v^5+48 u^4 v^5+2112 u^5 v^5-3072 u^6 v^5-3 u v^6+75 u^2 v^6

-96 u^3 v^6-384 u^4 v^6$$
 * $$-3072 u^5 v^6+4096 u^6 v^6-3 u v^7-24 u^2 v^7-

128 u^3 v^7+v^8=0$$