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Nathan Carter Senior Seminar Project Spring 2012.

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Presentation on theme: "Nathan Carter Senior Seminar Project Spring 2012."— Presentation transcript:

1 Nathan Carter Senior Seminar Project Spring 2012

2 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.

3 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.

4 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.

5 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.

6 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.

7 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.

8 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.

9 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.”

10 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.

11 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.

12 Works Cited

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