Editing Addition theorem

{{short description|Result that expresses a function f(x + y) in terms of f(x) and f(y)}}
{{About|addition theorems in general|specific addition theorems for trigonometric functions|angle addition formulas}}
In [[mathematics]], an '''addition theorem''' is a formula such as that for the [[exponential function]]:

:''e''<sup>''x''&thinsp;+&thinsp;''y''</sup> = ''e''<sup>''x''</sup>&thinsp;·&thinsp;''e''<sup>''y''</sup>,

that expresses, for a particular [[function (mathematics)|function]] ''f'', ''f''(''x''&thinsp;+&thinsp;''y'') in terms of ''f''(''x'') and ''f''(''y''). Slightly more generally, as is the case with the [[trigonometric functions]] {{math|sin}} and {{math|cos}}, several functions may be involved; this is more apparent than real, in that case, since there {{math|cos}} is an [[algebraic function]] of {{math|sin}} (in other words, we usually take their functions both as defined on the [[unit circle]]).

The scope of the idea of an addition theorem was fully explored in the nineteenth century, prompted by the discovery of the addition theorem for [[elliptic function]]s. To "classify" addition theorems it is necessary to put some restriction on the type of function ''G'' admitted, such that

:''F''(''x''&thinsp;+&thinsp;''y'') = ''G''(''F''(''x''), ''F''(''y'')).

In this identity one can assume that ''F'' and ''G'' are vector-valued (have several components). An '''algebraic addition theorem''' is one in which ''G'' can be taken to be a vector of [[polynomial]]s, in some set of variables. The conclusion of the mathematicians of the time was that the theory of [[abelian function]]s essentially exhausted the interesting possibilities: considered as a [[functional equation]] to be solved with polynomials, or indeed [[rational function]]s or [[algebraic function]]s, there were no further types of solution.

In more contemporary language this appears as part of the theory of [[algebraic group]]s, dealing with [[Algebraic_group#Glossary_of_algebraic_groups|commutative]] groups. The connected, [[projective variety]] examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an [[abelian variety]] by rather weak conditions on its group law. The so-called [[quasi-abelian function]]s are all known to come from extensions of abelian varieties by commutative affine group varieties. Therefore, the old conclusions about the scope of global algebraic addition theorems can be said to hold. A more modern aspect is the theory of [[formal group]]s.

==See also==
*[[Timeline of abelian varieties]]
*[[Spherical harmonics#Addition theorem|Addition theorem for spherical harmonics]]
*[[Mordell–Weil theorem]]

==References==
*{{Springer|id=A/a110350|title=Addition theorems in the theory of special functions}}

{{DEFAULTSORT:Addition Theorem}}
[[Category:Theorems in algebraic geometry]]
[[Category:Theorems in algebra]]