The first lemniscate constant is given by
Todd (1975) refers to a proof by T. Schneider that this is a transcendental number. The general lemniscatic case is
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
Since , is in a hyperbolic region. In the complete case () (19.20.14) reduces to