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Nathan Carter Senior Seminar Project Spring 2012

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Some History of Cauchy Cauchy lived from August 21, 1789 to May 23, 1857. Cauchy spent most of his years as a mathematician in France. Cauchy was a French mathematician who found interest in analysis. He also came up with proofs for the theorems of infinitesimal calculus. Cauchy was a big contributor to group theory in abstract algebra.

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Cauchy Was Influenced by a Few Mathematicians Lagrange, for instance, gave Cauchy a problem. This problem that Lagrange gave Cauchy marked the beginning of Cauchy’s mathematical career. Lagrange’s problem that Cauchy had to solve was for Cauchy to figure out whether the angles of a convex polyhedron are determined by its faces.

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Cauchy’s Future Endeavors Cauchy had a bright future ahead of him. He discovered many different types of formulas and theorems that mathematicians still use widely today. Three important examples of Cauchy’s discoveries are the Cauchy sequence, the Cauchy integral formula, and the Cauchy mean value theorem.

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Cauchy’s Sequence: Mainly Pertains to Analysis Cauchy derived a sequence that is very intriguing to people who are interested in Mathematics. The sequence that Cauchy derived can be defined as a sequence in which the elements of that sequence tend to close in on one another as the same sequence progresses.

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Graphs of Cauchy’s Sequence vs. Non-Cauchy Sequence Here is a contrast between a Cauchy sequence and a non Cauchy sequence. This is a Cauchy sequence.This is a non Cauchy sequence.

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Cauchy’s Integral Formula Cauchy’s integral formula is used widely for Complex Analysis. Cauchy used his integral formula to make it clear that differentiation of a function is identical to the integration of that same function. That is, taking the integration of a function is the same as solving for a differential equation.

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Cauchy’s Integral Formula Continued Cauchy used his integral formula to show how many times a certain object travels around the circumference of a circle.

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Cauchy Described a Theorem about Group Theory Cauchy’s theorem is as follows: “If G is a finite group and p is a prime number that divides the order of G, also known as the number of elements in G, then G contains an element of order p.”

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Cauchy’s Theorem (Group Theory) Continued So, there exists an element z that belongs to G where p is the lowest value of the elements contained in G. Note that G is a finite group. It is sufficient to say that p is a non-zero value. Therefore, if you take z*z*z all the way to a prime number of p times, your result is z p = e. The element e is also known as the identity element. Thus, any value in the finite group G that is combined with the element e in G will return that same value as a result.

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Cauchy’s Mean Value Theorem Cauchy’s mean value theorem is widely used in mathematical analysis. Cauchy’s mean value theorem says that f(x) and g(x) are continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Also, observe that g(a) and g(b) must not be equal to each other. Thus, there exists a value c where a < c < b such that the following formula is true.

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Works Cited http://en.wikipedia.org/wiki/Main_Page http://www.britannica.com http://www.google.com/imghp http://www.math.berkeley.edu http://www.math.psu.edu http://www.thefreedictionary.com http://www.wolfram.com/

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