# §19.20 Special Cases

## §19.20(i)

In this subsection, and also §§19.20(ii)19.20(v), the variables of all -functions satisfy the constraints specified in §19.16(i) unless other conditions are stated.

The first lemniscate constant is given by

19.20.2

Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. The general lemniscatic case is

## §19.20(iii)

19.20.7 or ; .

where may be permuted.

When the variables are real and distinct, the various cases of are called circular (hyperbolic) cases if is positive (negative), because they typically occur in conjunction with inverse circular (hyperbolic) functions. Cases encountered in dynamical problems are usually circular; hyperbolic cases include Cauchy principal values. If are permuted so that , then the Cauchy principal value of is given by

valid when

19.20.15

or

19.20.16

Since , is in a hyperbolic region. In the complete case () (19.20.14) reduces to

## §19.20(iv)

The second lemniscate constant is given by

19.20.22

Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. Compare (19.20.2). The general lemniscatic case is

## §19.20(v)

Define . Then

where is defined by (19.19.1). Also,