Extensions and Applications
The field can be algebraically closed by adjoining an imaginary unit (i), or by letting the coefficients be complex. It is rich enough to allow a significant amount of analysis to be done, but its elements can still be represented on a computer in the same sense that real numbers can be represented using floating point. It is the basis of automatic differentiation, a way to perform differentiation in cases that are intractable by symbolic differentiation or finite-difference methods.
Hahn series (with real coefficients and value group ) are a larger field which relaxes the condition on the support of being left finite to that of being well-ordered (i.e., admitting no infinite decreasing sequence): this gives a meaning to series such as which are not in the Levi-Civita field.
Read more about this topic: Levi-Civita Field
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