In his 1758 book, Lambert (1728–1777) established the most compact expression of refraction integral, and developed it in a power series sine or tangent zenithal distance. He then examined geodetic consequences curvature light rays.

An exact transformation, which we call the master identity, is obtained for first time series $$\sum _{n=1}^{\infty }\sigma _{a}(n)e^{-ny}$$ $$a\in {\mathbb {C}}$$ and Re $$(y)>0$$ . New modular-type transformations when a nonzero even integer are as its special cases. The precise obstruction to modularity explicitly seen in these transformations. These include novel companion Ramanujan’s famou...

Adaptive composite map projections transform projections as the user changes the scale of the web map. The equal-area version of the original adaptive map projection composite uses the Lambert azimuthal equal-area projection for regional maps, and three equal-area world projections, namely the Hammer, the Eckert-Greifendorff, and the quartic authalic projections. These three projections show po...