Definition:Cauchy Product
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Definition
Let $A := \ds \sum_{n \mathop = 0}^\infty a_n$ and $B := \ds \sum_{n \mathop = 0}^\infty b_n$ be two series.
The Cauchy product $C$ of $A$ and $B$ is defined as:
- $\ds C := \sum_{n \mathop = 0}^\infty c_n = \sum_{n \mathop = 0}^\infty a_n \cdot \sum_{n \mathop = 0}^\infty b_n$
where:
- $\ds \forall n \in \N: c_n = \sum_{k \mathop = 0}^n a_k b_{n - k} = \sum_{k \mathop = 0}^n a_{n - k} b_k$
Source of Name
This entry was named for Augustin Louis Cauchy.
Sources
- 1992: Larry C. Andrews: Special Functions of Mathematics for Engineers (2nd ed.) ... (previous) ... (next): $\S 1.2.3$: Operations with series