Non User
Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z	Business Non User Industries Finance Tax

Home > Cauchy product

is { c₁ }_n=1^∞ where

1 Formal power series

This concept is captures the essence of multiplication of formal power series. Given two such series,

and

their product is

One multiplies just as if one were working with algebra of finite sums without worrying about questions of convergence. It has the same "form" as multiplication of finite sums, and is therefore called "formal" multiplication.

2 Convergence

If one works with convergent power series rather than formal power series, does Cauchy multiplication give correct results, i.e., does the product of the two series converge, and is the scalar to which it converges equal to the product of the sums of the other two series? A partial answer is this: If one series of complex numbers converges, and the other converges absolutely, then the answer to both questions is "yes".

3 A variant

Often one writes formal power series in the form

so that the coefficients may be thought of as values of the derivative of the formal power series at 0. (However, since formal power series are not in general convergent power series, to speak of their "values" may be problematic.)

The product of that series with

is

where

One may see this relation written in the form

and some writers on the subject may call the resulting sequence { c_n }_n=1^∞ the Cauchy product.

Politics	Business	Finance

Topics: Cauchy Product Series Power Formal Multiplication Infin Sums...

---

Cauchy sequenceIn mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses. They are named after the French mathematician Augustin Louis Cauchy. They are of interest because, given certain condi Cauchy distributionThe Cauchy distribution is a probability distribution with probability density function : where x is the location parameter and s is the scale parameter''. The special case when x 0 and s 1 is called the standard Cauchy distribution with the probability d Cauchy-Riemann equationsPartial differential equations Complex analysis In complex analysis, the Cauchy-Riemann differential equations are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. Let f u + iv be a fu Cauchy-Schwarz inequalityIn mathematics, the Cauchy-Schwarz inequality also known as the Schwarz inequality or the Cauchy-Bunyakovski-Schwarz inequality is a useful inequality encountered in many different settings, such as linear algebra applied to vectors, in analysis applied t Cauchy integral theoremComplex analysis Theorems The Cauchy integral theorem in complex analysis is an important statement about path integrals for holomorphic functions in the complex plane. Essentially, it says that if two different paths connect the same two points, and a fu Cauchy's integral formulaCauchy's integral formula is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk. It can also be used to formulate integral formula Cauchy-Binet formula Cauchy-à-la-Tour Cauchy principal value Cauchy horizon Cauchy product Cauchy (crater) Cauchy space Cauchy-Euler equation