Cauchy's Integral Theorem on a Finitely Generated, Real, Commutative, and Associative Algebra
✍ Scribed by Paul S. Pedersen
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 317 KB
- Volume
- 131
- Category
- Article
- ISSN
- 0001-8708
No coin nor oath required. For personal study only.
✦ Synopsis
Let R[:]=R[: 1 , : 2 , ..., : n ] (where : 1 =1) be a real, unitary, finitely generated, commutative, and associative algebra. We consider functions
We impose a total order on an algorithmically defined basis B for R[:]. The resulting algebra and ordered basis will be written as (R[:], <). We then use this basis to define a norm &} & on (R[:], <). Continuous functions, differentiable functions, and the concept of Riemann integration will then be defined and discussed in this new setting. We then show that # f (z) dz=0 when f (z) is a continuous and differentiable function defined in a simply connected region G/R[:] n /(R[:], <) containing the closed path #.