Periodic nodal surfaces and exponential scale

Abstract

Abstract Both, Periodic Nodal Surfaces (PNS) and exponential scale surfaces (ES), are suitable tools to investigate and describe the organisation of crystal structures. Therefore the mathematical relationship between ES and PNS is investigated. It is shown that the function cos (x) within the range 0 ≤ x ≤ 2π can be developed into a series in terms of orthogonal functions Φ n (x) = Σ k = 1 n c n k exp (cos ((k–1)x)). An orthonormal set of functions Φ n (x) has been evaluated using Gram-Schmidt orthonormalization. Series representation of cos (x) and sin (x) in terms of exp (sin ((k–1)x)) is impossible.