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MathematicalPhysicist
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how does weierstrass' approach to analysis differ from the classical approach, i.e leibnizs' and Newton's calculus?
I said that Weierstrass SYSTEMATIZED the use of this technique; I didn't say he was the the first to have these ideas.quasar987 said:And how does Cauchy fit in that picture? I thought he was the one who invented the epsilon-delta formulation
from what i read his approach had left behind the infinitisemals for the the epsilon-delta formulation, but i also read that abraham robinson had resurrected the infinitisimal formulation on a better rigorous formulation, in what now is called Non-Standard Analysis (NSA), how do these approaches differ from each other, and what makes NSA non-standard?arildno said:Since Weierstrass was the one systematizing the "epsilon/delta"-approach to calculus, you might say that Weierstrass was the first to provide a truly mature and rigorous approach to calculus.
I hope math-wizzes like Hurkyl, M.G, or mathwonk can give you a bit of solid info on robinson's approach, but here's a few schematic details on the history of analysis that I don't think is too misleading:loop quantum gravity said:from what i read his approach had left behind the infinitisemals for the the epsilon-delta formulation, but i also read that abraham robinson had resurrected the infinitisimal formulation on a better rigorous formulation, in what now is called Non-Standard Analysis (NSA), how do these approaches differ from each other, and what makes NSA non-standard?
loop quantum gravity said:from what i read his approach had left behind the infinitisemals for the the epsilon-delta formulation, but i also read that abraham robinson had resurrected the infinitisimal formulation on a better rigorous formulation, in what now is called Non-Standard Analysis (NSA), how do these approaches differ from each other, and what makes NSA non-standard?
The Weierstrass approach to analysis is a method used in mathematical analysis to prove the existence of a continuous function on a closed interval. It was developed by German mathematician Karl Weierstrass in the 19th century.
The Weierstrass approach differs from other approaches to analysis in that it focuses on constructing a continuous function using a series of simpler functions, rather than directly proving the existence of a continuous function. This allows for more flexibility and makes it easier to prove the existence of a continuous function in more complex cases.
The main idea behind the Weierstrass approach is that any continuous function can be approximated by a sequence of simpler functions, such as polynomials or trigonometric functions. By using these simpler functions, we can construct a continuous function that closely resembles the original function.
The Weierstrass approach has many applications in mathematics, particularly in real analysis and the study of function spaces. It is used to prove the existence of continuous functions in various mathematical theorems, such as the intermediate value theorem and the Stone-Weierstrass theorem. It is also used in the construction of fractals and in the study of Fourier series.
While the Weierstrass approach is a powerful tool in mathematical analysis, it does have some limitations. It can only be used to prove the existence of a continuous function, not its uniqueness. Additionally, it can be difficult to determine the rate of convergence of the series of simpler functions, leading to potential errors in the approximation of the original function.