§23.5 Special Lattices

§23.5(i) Real-Valued Functions

The Weierstrass functions take real values on the real axis iff the lattice is fixed under complex conjugation: ; equivalently, when . This happens in the cases treated in the following four subsections.

§23.5(ii) Rectangular Lattice

This occurs when both and are real and positive. Then and the parallelogram with vertices at 0, , , is a rectangle.

In this case the lattice roots , , and are real and distinct. When they are identified as in (23.3.9)

23.5.1

Also, and have opposite signs unless , in which event both are zero.

As functions of , and are decreasing and is increasing.

§23.5(iii) Lemniscatic Lattice

This occurs when is real and positive and . The parallelogram 0, , , is a square, and

23.5.2
23.5.3

Note also that in this case . In consequence,

§23.5(iv) Rhombic Lattice

This occurs when is real and positive, , , and . The parallelogram 0, , , , is a rhombus: see Figure 23.5.1.

The lattice root is real, and , with . and have the same sign unless when both are zero: the pseudo-lemniscatic case. As a function of the root is increasing. For the case see §23.5(v).

§23.5(v) Equianharmonic Lattice

This occurs when is real and positive and . The rhombus 0, , , can be regarded as the union of two equilateral triangles: see Figure 23.5.2.

 Figure 23.5.1: Rhombic lattice. . Symbols: : real part and , , : lattice generators Referenced by: §23.5(iv) Permalink: http://dlmf.nist.gov/23.5.F1 Encodings: pdf, png Figure 23.5.2: Equianharmonic lattice. , . Symbols: : base of exponential function and , , : lattice generators Referenced by: §23.5(v) Permalink: http://dlmf.nist.gov/23.5.F2 Encodings: pdf, png

and the lattice roots and invariants are given by