Heuman Lambda Function

$$\Lambda_0(\phi\vert m)\equiv {F(\phi\vert 1-m)\over K(1-m)}+{2\over\pi}K(m)Z(\phi\vert 1-m),$$

where $\phi$ is the Amplitude, $m$ is the Parameter, $Z$ is the Jacobi Zeta Function, and $F(\phi\vert m')$ and $K(m)$ are incomplete and complete Elliptic Integrals of the First Kind.

See also Elliptic Integral of the First Kind, Jacobi Zeta Function

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 595, 1972.