The discussion revolves around the interpretation of the phrase "zero to order" in the context of contour integrals of analytic functions. Participants explore its meaning in relation to power series expansions and the implications for coefficients in such expansions.
Participants express differing interpretations of what "zero to order" entails, particularly regarding the coefficients in the power series expansion. There is no consensus on the implications of this terminology.
The discussion highlights ambiguity in the terminology used and the lack of formal definitions, which may lead to different interpretations among participants.
It means that if you were to write the value of the integral as a power series function of e, a0+a1e+a2e2+a3e3+... then the a0, a1 and a2 coefficients would be zero.Tomtam said:I saw the sentence " So the contour integral of an analytic function f(z) around a tiny square of size e is zero to order e^2. ". I want to know what " be zero to order " means exactly.
Perhaps, but I don't think so. Expressing a function to second order means taking the expansion terms up to and including the x2 term. If it is "zero to second order" that should mean the second order approximation is still zero.FactChecker said:I think that a2 can be non-zero.
I stand corrected. I think you are probably right. I was thinking of a second order zero, but the phrase "zero to second order" does sound more like your definition. I don't think I have ever heard that terminology formally defined or used.haruspex said:Perhaps, but I don't think so. Expressing a function to second order means taking the expansion terms up to and including the x2 term. If it is "zero to second order" that should mean the second order approximation is still zero.