mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an addition theorem is a formula such as that for the exponential function: :''e''^{''x'' + ''y''} = ''e''^''x'' · ''e''^''y'', that expresses, for a particular

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

''f'', ''f''(''x'' + ''y'') in terms of ''f''(''x'') and ''f''(''y''). Slightly more generally, as is the case with the

trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

and , several functions may be involved; this is more apparent than real, in that case, since there is an

algebraic function In mathematics, an algebraic function is a function that can be defined as the root of an irreducible polynomial equation. Algebraic functions are often algebraic expressions using a finite number of terms, involving only the algebraic operati ...

of (in other words, we usually take their functions both as defined on the

unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...

). 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 In the mathematical field of complex analysis, elliptic functions are special kinds of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Those integrals are ...

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 of

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s, in some set of variables. The conclusion of the mathematicians of the time was that the theory of abelian functions essentially exhausted the interesting possibilities: considered as a

functional equation In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...

to be solved with polynomials, or indeed

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

s or

s, there were no further types of solution. In more contemporary language this appears as part of the theory of

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

s, dealing with

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

groups. The connected,

projective variety In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...

examples are indeed exhausted by abelian functions, as is shown by a number of results characterising an

abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group ...

by rather weak conditions on its group law. The so-called quasi-abelian functions 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 In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one o ...

References

* {{DEFAULTSORT:Addition Theorem Theorems in algebraic geometry Theorems in algebra

See also

References