What does ".... is zero to order ...." mean?

[Tomtam]

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.

 

[haruspex]

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.

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.

 

[FactChecker]

I think that a₂ can be non-zero.

 

[haruspex]

I think that a₂ can be non-zero.

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]

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.

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.