It is well known that Cauchy was the first to define the derivative of a function in terms of a rigorous definition of limit. Even more important, he used his definitions to prove theorems about the derivative.
So, what do we mean by the derivative of a complex function? Analogous to the definition of the derivative for real functions, we can write: Substituting for u + iv for f(x) and taking the limit along the real axis gives: Now, we group the real parts and the imaginary parts: ...
In 1826[6] Cauchy gave a formal definition of a residue of a function. This concept regards functions that have poles—isolated singularities, i.e., points where a function goes to infinity. If the complex-valued function f(z) can be expanded in the neighborhood of a singularity a as whe...
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4) These ε-δ Cauchy’s proof method became the standard definition ofLimit of Functionin Analysis. 5) They are notorious for causing widespread discomfort among future math students. In fact, when it was first introduced by Cauchy in theEcole PolytechniqueLecture, the French Napoleon top studen...
英文: Fortunately, we have contrived mathematics in which the definition of limit is given by great mathematicians Louis Cauchy and Weierstrass, and what surprises us is that the demon has a seraphic appearance. 中文: 这个“魔鬼”无处不在,数学家们历经千年的磨砺,直到19世纪柯西和魏尔斯特拉斯给出...
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is a convex set, and (b) that the set of points C={x:‖x‖=1} is a nonconvex one. 8.3 Show the first-order convexity condition. 8.4 Show that a function f is convex if the one-dimensional function g(t):=f(x+ty) is convex, ∀x,y in the domain of definition of f. 8.5...
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2) Cauchy mean value function Cauchy中值函数 1. Based on this, the definition of Cauchy mean value function is given in this paper. 文献[2～6]对微分中值定理“中间点”的渐近性质进行了研究,本文在此基础上,给出了“Cauchy中值函数”的定义,对Cauchy中值函数的分析性质进行了系统的综合讨论,证明了...