where
D = 1 − k2s12s22.
The introduction of the function ƒ1(z) is equivalent to the introduction of the function ℜ(z; ω, 2ω′) constructed from the periods ω, 2ω′ as was ℜ(z) from ω and ω′; denoting this function by ℜ1(z) and its differential coefficient by ℜ′1(z), we have in fact
| ƒ1(z) = ½ | ℜ′1(z) |
| ℜ1(ω′) − ℜ1(z) |
as we see at once by considering the zeros and poles and the limit of zƒ1(z) when z = 0. In terms of the function ℜ1(z) the original function ℜ(z) is expressed by
ℜ(z) = ℜ1(z) + ℜ1(z + ω′) − ℜ1(ω′),
as a consideration of the poles and expansion near z = 0 will show.
A function having ω, ω′ for periods, with poles at two arbitrary points a, b and zeros at a′, b′, where a′ + b′ = a + b save for an expression mω + m′ω′, in which m, m′ are integers, is a constant multiple of
| {ℜ [z − ½(a′ + b′)] − ℜ [a′ − ½(a′ + b′)]} / {ℜ [z − ½(a + b)] − ℜ |
if the expansion of this function near z = a be