The phrase "zero to order e^2" in the context of contour integrals indicates that the coefficients a0, a1, and a2 in the power series expansion of the integral's value are all zero. This means that the integral's approximation does not include terms up to the second order, confirming that the second order approximation remains zero. The discussion clarifies that while there may be confusion regarding the terminology, the consensus is that "zero to second order" implies the absence of non-zero contributions up to that order.
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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.