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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
\end{displaymath} (1)

\mathop{\rm arccoshlemn}x\equiv \int_x^1 (1+t^4)^{1/2}\,dt.
\end{displaymath} (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}},
\end{displaymath} (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),
\end{displaymath} (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)},
\end{displaymath} (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]}
\end{displaymath} (6)


\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]}
\end{displaymath} (7)

(Berndt 1994).

See also Lemniscate Function


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

© 1996-9 Eric W. Weisstein