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.
"Zero to order" refers to a mathematical concept in which a function or series is approximated by setting the first term to a value of zero. This allows for a simplified calculation of the function's behavior without considering higher order terms.
Scientists use the concept of "zero to order" to simplify complex calculations and model the behavior of functions or series. It allows for a quick approximation and can provide valuable insights into the overall behavior of a system.
One example of "zero to order" is in Taylor series, where the first term is set to zero to approximate the function's behavior at a certain point. This allows for a simplified calculation of the function's derivatives and can be used to model physical phenomena such as motion or heat transfer.
Unlike other approximations that use higher order terms to improve accuracy, "zero to order" focuses on only the first term in the series. This can result in a less accurate approximation, but it is often sufficient for understanding the overall behavior of a function.
Yes, "zero to order" is a simplified approximation and may not accurately represent the behavior of a function in all cases. It should be used with caution and in combination with other mathematical techniques to ensure accurate results.