An algebraic addition theorem for a nonconstant analytic function 
is an identity
 |
(1)
|
where 
is a nonzero polynomial with constant complex coefficients. The identity
is required to hold for all 
and 
in a neighborhood where the
three values are defined. It is a special kind of argument
addition relation, and need not determine 
uniquely (Hancock 1910, pp. 33-34).
Writing 
,
,
and 
,
the identity function has the relation
 |
(2)
|
while the exponential function has
 |
(3)
|
For the tangent, the trigonometric addition formulas give
 |
(4)
|
The cosine and sine respectively satisfy
 |
(5)
|
and
 |
(6)
|
These last two relations involve only the named trigonometric function, with no companion sine or cosine.
They can be obtained from the trigonometric
addition formulas and the Pythagorean identity
by eliminating companion functions, for example using a Gröbner
basis (Trott 2006). The relations hold wherever the function values are finite.
They are quadratic and quartic in 
for the cosine and sine,
respectively, so a branch must still be selected when
solving for 
.
Among nonconstant functions meromorphic on the whole complex plane, those admitting an algebraic
addition theorem are precisely rational functions,
rational functions of 
for a nonzero complex number 
,
and elliptic functions (Hancock 1910, p. 457).
