Hyperbolic Lemniscate Function

By analogy with the Lemniscate Functions, hyperbolic lemniscate functions can also be defined

$$\mathop{\rm arcsinhlemn}x\equiv \int_0^x (1+t^4)^{1/2}\,dt$$ (1)

$$\mathop{\rm arccoshlemn}x\equiv \int_x^1 (1+t^4)^{1/2}\,dt.$$ (2)

Let $0\leq\theta\leq\pi/2$ and $0\leq v\leq 1$, and write

$${\theta\mu\over 2}=\int_0^v {dt\over\sqrt{1+t^2}},$$ (3)

where $\mu$ is the constant obtained by setting $\theta=\pi/2$ and $v=1$. Then

$$\mu={2\over\pi} K\left({1\over\sqrt{2}\,}\right),$$ (4)

where $K(k)$ is a complete Elliptic Integral of the First Kind, and Ramanujan showed

$$2\tan^{-1} v=\theta+\sum_{n=1}^\infty {\sin(2n\theta)\over n\cosh(n\pi)},$$ (5)

$${\textstyle{1\over 8}}\pi-{\textstyle{1\over 2}}\tan^{-1}(v^... ...2n+1)\theta]\over(2n+1)\cosh[{\textstyle{1\over 2}}(2n+1)\pi]}$$ (6)

and

$$\ln\left({1+v\over 1-v}\right)=\ln[\tan({\textstyle{1\over 4... ...\infty {(-1)^n\sin[(2n+1)\theta]\over (2n+1)[e^{(2n+1)\pi}-1]}$$ (7)

(Berndt 1994).

See also Lemniscate Function

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 255-258, 1994.