- Addition theorem
In

mathematics , an**addition theorem**is a formula such as that for theexponential function :"e"

^{"x" + "y"}= "e"^{"x"}·"e"^{"y"}that expresses, for a particular function "f", "f"("x" + "y") in terms of "f"("x") and "f"("y"). Slightly more generally, as is the case with the

trigonometric function s "sin" and "cos", several functions may be involved; this is more apparent than real, in that case, since there "cos" is analgebraic function of "sin" (in other words, we usually take their functions both as defined on theunit 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 forelliptic function s. To 'classify' addition theorems it is necessary to put some restriction on the type of function "G" admitted, such that:"F"("x" + "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 ofpolynomial s, in some set of variables. The conclusion of the mathematicians of the time was that the theory ofabelian function s essentially exhausted the interesting possibilities: considered as afunctional equation to be solved with polynomials, or indeedrational function s oralgebraic 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 commutative groups. The connected,projective variety examples are indeed exhausted by abelian functions, as is shown by a number of results characterising anabelian variety by rather weak conditions on its group law. The so-calledquasi-abelian function s are known all 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 offormal group s.

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