Pierre alphonse laurent biography samples

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Either write something worth reading or do something worth writing.
— Benjamin Franklin (1706-1790)
Math and music are two areas that have attracted my attention ever since I was in elementary and high school. No doubt, a couple of teachers influenced me, and for sure my father did too as he often spoke with admiration of the great mathematicians and musicians. Besides, the two areas are linked: both show the beauties of striking and magnificent consonances and dissonances. Thereafter, already at the engineering school of the University of Buenos Aires, another well-versed mathematician, Juan Carlos Vignaux (1893–1984), led me with enthusiasm into deeper and more powerful concepts and tools. Convolution soon caught my interest and made me smile with the pleasing description of a visiting guest function that turns around and greets the host function. Isn’t that a beauty? Not long after, many times I had to use the operation as a mere applied electronics engineer calculating transfer functions, often bouncing my head against a wall when the integrals became too thorny or when the Laplace transform table did not consider the fancy function for which I was looking.
I never bothered much to find out who was the owner of the brain that left us with such a superb inheritance until, many years later, the history of science and technology began to call my wits. Since the IEEE Pulse “Retrospectroscope” column was placed in my hands by its kind editors back in 2011, I have been like a hunter in a permanent search for adequate subjects to dig into their respective pasts, and convolution returned to my memory. Swiftly and with pleasure, I started to probe into my files, old notes (I even called a former colleague, a math professor), and ended up navigating through our all-mighty Internet. At one point, and already having a coarse preliminary draft, I ran across th

Power series with negative powers

This article is about doubly infinite power series. For power series with finitely many negative exponents, see Formal Laurent series.

In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894.

Definition

The Laurent series for a complex function about an arbitrary point is given by where the coefficients are defined by a contour integral that generalizes Cauchy's integral formula:

The path of integration is counterclockwise around a Jordan curve enclosing and lying in an annulus in which is holomorphic (analytic). The expansion for will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled . When is defined as the circle, where , this amounts to computing the complex Fourier coefficients of the restriction of to . The fact that these integrals are unchanged by a deformation of the contour is an immediate consequence of Green's theorem.

One may also obtain the Laurent series for a complex function at . However, this is the same as when .

In practice, the above integral formula may not offer the most practical method for computing the coefficients for a given function ; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is unique whenever it exists, any expression of this form that equals the given function in some annulus must actually be the Laurent expansion of .

Convergence

Laurent ser

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Biography

Augustin-Louis Cauchy (Paris, August 21, 1789 – Sceaux, May 23, 1857) was one of the most prominent mathematicians of the first half of the nineteenth century. He was the first to give a rigorous basis to the concept of limit. His criterion for the convergence of sequences defines sequences that are now known as Cauchy sequences. This notion has led to the fundamental mathematical concept of a complete space. The Cauchy condition for the convergence of series can be found in any present-day textbook on calculus. Probably Cauchy is most famous for his single-handed development of complex function theory, with Cauchy's residue theorem as the fundamental result.

Cauchy was a prolific writer, he wrote approximately eight hundred research articles and five complete textbooks. He was a devout Roman Catholic, strict (Bourbon) royalist, and a close associate of the Jesuit order.

Biography

Youth and education

Cauchy's father (Louis-François Cauchy) was a high official in the Parisian Police of the Old Régime. He lost his position because of the French Revolution (July 14, 1789) that broke out one month before Augustin-Louis was born. The Cauchy family survived the revolution and the following Reign of Terror (1794) by escaping to Arcueil, where Cauchy received his first education, from his father. After the death of Robespierre (1794), it was safe for the family to return to Paris. There Louis-François Cauchy found himself a new bureaucratic job, and quickly moved up the ranks. When Napoleon Bonaparte came to power (1799), Louis-François Cauchy was further promoted, and became Secretary-General of the Senate, working directly under Laplace (who is now better known for his work on mathematical physics). The famous mathematician Lagrange was also no stranger i