The inverse nome is obtained by solving the nome 
for the parameter 
and is given by
 |  |  |
(1)
|
 |  |  |
(2)
|
for 
, where 
is a Jacobi
theta function, 
is the Dedekind eta function, and 
is the nome.
The inverse nome function is essentially the same as the elliptic lambda function, the difference being that elliptic
lambda function is a function of the half-period
ratio 
,
while the inverse nome is a function of the nome 
, where 
is itself a function of 
.
The inverse nome is implemented as InverseEllipticNomeQ[q] in the Wolfram Language.
For a real number 
with 
, the inverse nome and nome
are mutual inverse functions. For a general complex number 
with 
, the principal branch
of the nome need not recover the original 
from 
. This reflects the modular
transformations under which the elliptic
lambda function is invariant (NIST 2026, §23.18).
Special values include
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
where the last row is a limit along the real axis. The series defining the Jacobi
theta functions converge strictly inside the unit
circle, which is a natural boundary with
a dense set of singularities. The last row does not
assign a value at 
.
It has series
 |
(6)
|
(OEIS A115977).
It satisfies
 |
(7)
|
