In mathematics, an addition theorem is a formula such as that for the exponential function
- ex + y = exยทey
that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) Slightly more generally, as is the case with the trigonometric functions sin and cos, several functions may be involved; this is more apparent than real, in that case, since there cos is an algebraic function of sin (in other words, we usually take their functions both as defined on the unit circle). In Mathematics, an algebraic function is informally a function which satisfies a Polynomial equation whose coefficients are themselves polynomials In Mathematics, a unit circle is
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 functions. The 19th century of the Common Era began on January 1, 1801 and ended on December 31, 1900, according to the Gregorian calendar In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic 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 polynomials, in some set of variables. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations 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 to be solved with polynomials, or indeed rational functions or algebraic functions, there were no further types of solution. In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety In Mathematics or its applications a functional equation is an Equation expressing a relation between the value of a function (or functions at a point with its values In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In In Mathematics, an algebraic function is informally a function which satisfies a Polynomial equation whose coefficients are themselves polynomials
In more contemporary language this appears as part of the theory of algebraic groups, dealing with commutative groups. In Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse 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. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety The so-called quasi-abelian functions 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 of formal groups. In Mathematics, a formal group law is (roughly speaking a Formal power series behaving as if it were the product of a Lie group.
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