In mathematics, the Cauchy product, named in honor of Augustin Louis Cauchy, of two strictly formal (not necessarily convergent) series Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... In mathematics, a series is often represented as the sum of a sequence of terms. ...

usually, of real or complex numbers, is defined by a discrete convolution as follows. The Cauchy product is In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

for n = 0, 1, 2, ...

"Formal" means we are manipulating series in disregard of any questions of convergence. These need not be convergent series. See in particular formal power series. In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is...

One hopes, by analogy with finite sums, that in cases in which the two series do actually converge, the sum of the infinite series In mathematics, a series is a sum of a sequence of terms. ...

is equal to the product

just as would work when each of the two sums being multiplied has only finitely many terms.

In sufficiently well-behaved cases, this works. But—and this is an important point—the Cauchy product of two sequences exists even when either or both of the corresponding infinite series fails to converge.

Examples

Finite series

x_i = 0 for all i > n and y_i = 0 for all i > m. Here the Cauchy product of and is readily verified to be . Therefore, for finite series (which are finite sums), Cauchy multiplication is direct multiplication of those series.

Infinite series

• For some , let and . Then

by definition and the binomial formula. Since, formally, and , we have shown that . Since the limit of the Cauchy product of two absolutely convergent series is equal to the product of the limits of those series (see below), we have proven the formula exp(a + b) = exp(a)exp(b) for all . In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, the Cauchy product, named in honor of Augustin Louis Cauchy, of two strictly formal (not necessarily convergent) series usually, of real or complex numbers, is defined by a discrete convolution as follows. ...

• As a second example, let x(n) = 1 for all . Then C(x,x)(n) = n + 1 for all so the Cauchy product and it does not converge.

Convergence and Mertens' theorems

Let x', y be real sequences. It was proved by Franz Mertens that if the series converges to Y and the series converges absolutely to X then their Cauchy product converges to XY. It is not sufficient for both series to be conditionally convergent. For example, the sequence x_n = ( − 1)ⁿ / n generates a conditionally convergent series but the sequence C(x,x) does not converge to 0. Here is a proof. In mathematics, Mertens theorems are three results in number theory related to the density of prime numbers, and proved by Franz Mertens. ... Franz Mertens (March 20, 1840 - March 5, 1927) was a German mathematician. ... In mathematics, a series is the sum of the terms of a sequence of numbers. ... In mathematics, a series is a sum of a sequence of terms. ... In mathematics, a series is a sum of a sequence of terms. ...

Proof of Mertens' theorem

Let , and . Then by rearangement. So . Fix ε > 0. Since is absolutely convergent and is convergent then there exists an integer N such that for all and an integer M such that for all (since the series converges, the sequence must converge to 0). Also, there exists an integer L such that if then . Therefore,

for all integers n larger than N, M, and L. By the definition of convergence of a series . In mathematics, a series is the sum of the terms of a sequence of numbers. ...

Cesàro's theorem

If x and y are real sequences and and then

Generalizations

All of the foregoing applies to sequences in (complex numbers). The Cauchy product can be defined for series in the spaces (Euclidean spaces) where multiplication is the inner product. In this case, we have the result that if two series converge absolutely then their Cauchy product converges absolutely to the inner product of the limits. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...