Upload
storm
View
134
Download
0
Embed Size (px)
DESCRIPTION
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. - PowerPoint PPT Presentation
Citation preview
AUGUSTIN-LOUIS CAUCHY
Nathan CarterSenior Seminar Project
Spring 2012
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 manytimes a certain object travels around the circumferenceof 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/