Feb 2011

Relation between the regular and principal parts of the Laurent expansion

We have previously investigated the Taylor expansion for the case |x|>a, where the method used for obtaining the Taylor expansion was valid only in the real domain. In February 2011, we obtained a rather simple but satisfactory proof. However, its extreme simplicity suggests that it is quite possible that the proof had already been known.

Let w=f(z) be a function in the complex plane. Let it be regular in domain D, and let us assume that it can be expanded into a Laurent series.

Performing the variable transformation z-a=1/(u-a)

and considering

w=f(z)=g(u),

we note that the coefficient of the regular part of f(z) coincides with the coefficient of the principal part of g(u) after the variable transformation, and that the coefficient of the principal part of f(z) coincides with the coefficient of the regular part of g(u). The coefficient a_{-n} of f(z) corresponds to the n-th differential coefficient of g(u).

The proof will be presented later.

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