Nathan Carter Senior Seminar Project Spring 2012 Augustin-Louis Cauchy
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.
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.
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.
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.
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.
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.
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.
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.”
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 zp = 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.
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.
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/