I had to simplify it a bit for those who don't have a PhD but this is the basics:
Consider the generalized form of the Jacobian elliptic functions in the context of a doubly periodic lattice:
sn(u,k)2+cn(u,k)2=1anddn(u,k)2+k2⋅sn(u,k)2=1,\mathrm{sn}(u, k)^2 + \mathrm{cn}(u, k)^2 = 1 \quad \text{and} \quad \mathrm{dn}(u, k)^2 + k^2 \cdot \mathrm{sn}(u, k)^2 = 1,sn(u,k)2+cn(u,k)2=1anddn(u,k)2+k2⋅sn(u,k)2=1,
where sn(u,k)\mathrm{sn}(u, k)sn(u,k), cn(u,k)\mathrm{cn}(u, k)cn(u,k), and dn(u,k)\mathrm{dn}(u, k)dn(u,k) are solutions to the differential equation
(dsn(u,k)du)2=(1−sn(u,k)2)(1−k2sn(u,k)2).\left( \frac{d\mathrm{sn}(u, k)}{du} \right)^2 = (1 - \mathrm{sn}(u, k)^2)(1 - k^2 \mathrm{sn}(u, k)^2).(dudsn(u,k))2=(1−sn(u,k)2)(1−k2sn(u,k)2).
Now, combining this with the Fourier series expansion for complex exponential periodicities:
f(x)=∑n=−∞∞aneinωx,wherean=1T∫0Tf(x)e−inωx dx,f(x) = \sum_{n=-\infty}^\infty a_n e^{i n \omega x}, \quad \text{where} \quad a_n = \frac{1}{T} \int_0^T f(x) e^{-i n \omega x} \, dx,f(x)=n=−∞∑∞aneinωx,wherean=T1∫0Tf(x)e−inωxdx,
we can represent non-trivial periodic solutions in terms of modular transformations τ=ω2ω1\tau = \frac{\omega_2}{\omega_1}τ=ω1ω2, where ω1\omega_1ω1 and ω2\omega_2ω2 are the half-periods of the lattice satisfying ω1τ−ω2=2mπi\omega_1 \tau - \omega_2 = 2m\pi iω1τ−ω2=2mπi.
For hyperbolic analogs, consider the identity:
cosh2(x)−sinh2(x)=1,\cosh^2(x) - \sinh^2(x) = 1,cosh2(x)−sinh2(x)=1,
which directly corresponds to the imaginary projection of elliptic integrals:
∫0ϕdθ1−k2sin2(θ)=u,where u=am−1(ϕ,k).\int_0^{\phi} \frac{d\theta}{\sqrt{1 - k^2 \sin^2(\theta)}} = u, \quad \text{where } u = \mathrm{am}^{-1}(\phi, k).∫0ϕ1−k2sin2(θ)dθ=u,where u=am−1(ϕ,k).
Finally, invoking Ramanujan's Master Theorem, the qqq-series expansion of the Dedekind eta function:
η(τ)=eπiτ/12∏n=1∞(1−e2πinτ),\eta(\tau) = e^{\pi i \tau / 12} \prod_{n=1}^\infty (1 - e^{2 \pi i n \tau}),η(τ)=eπiτ/12n=1∏∞(1−e2πinτ),
where τ=ω2ω1\tau = \frac{\omega_2}{\omega_1}τ=ω1ω2 connects elliptic integrals with modular forms, providing a transcendental bridge between the realms of trigonometric and elliptic phenomena.