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

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1. Aug 12, 2017

### 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.

2. Aug 12, 2017

### haruspex

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.

3. Aug 20, 2017

### FactChecker

I think that a2 can be non-zero.

4. Aug 20, 2017

### haruspex

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.

5. Aug 21, 2017

### FactChecker

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.