ARITHMETIZATION OF ANALYSIS19TH Century Mathematics of Germany

Prepared by: Ma. Irene G. Gonzales

When the theory of a mathematical operation is poorly understood, what is likely to happen?

o Attracted by the powerful applicability ofanalysis, and lacking a real understanding ofthe foundations upon which the subject mustrest, mathematicians manipulated analyticalprocesses in an almost blind manner.

o A gradual accumulation of absurdities wasbound to result until some conscientiousmathematicians felt bound to attempt thedifficult task of establishing a rigorousfoundation under the subject.

FIELD OF ANALYSIS

Jean-le-Rond d’Alembert (1717 – 1783)

He gave the first suggestion of a real remedy for the unsatisfactory state of the foundations of analysis.

Joseph Louis Lagrange (1736 – 1813)

Joseph Louis Lagrange (1736 – 1813)

An Italian Mathematician

Earliest mathematician of the first rank to attempt a rigorization of the calculus.

His attempt was based upon representing a function by a Taylor’s series expansion.

His attempt was published in 1797 in Lagrange’s monumental work, Theorie des fonctions analytiques.

Carl Freidrich Gauss

Carl Freidrich Gauss

He broke from intuitive ideas and set newhigh standards of mathematical rigor.

The first adequate consideration of theconvergence of an infinite series wasencountered in Gauss’ treatment of thehypergeometric series.

Augustin-Louis Cauchy (1789 – 1857)

Augustin-Louis Cauchy (1789 – 1857)

A French Mathematician

He successfully executed d’Alembert’ssuggestion, by developing an acceptable theory of limits.

He defined continuity, differentiability, and the definite integral in terms of the limit concept.

Cauchy’s rigor inspired other mathematicians to join the effort to rid analysis of formalism and intuitionism.

Karl Theodor Wilhelm Weierstrass

Karl Theodor Wilhelm Weierstrass

a German Mathematician

He discovered the theoretical existence of a continuous

function having no derivative (in other words, a continuous

curve possessing no tangent at any of its points).

He saw the need for a rigorous “arithmetization” of calculus.

He advocated a program wherein the real number system itself

should be first rigorized, then all the basic concepts of analysis

should be derived from this number system.

Karl Theodor Wilhelm Weierstrass

And today all of analysis can be logically derived from a postulate set characterizing the real number system.

Along with Riemann andAugustin-Louis Cauchy, Weierstrass completely reformulated calculus in an even more rigorous fashion, leading to the development of mathematical analysis.

Bernhard Bolzano

Bernhard Bolzano

a Bohemian priest

one of the earliest mathematicians to begin instilling rigor into mathematical analysis

He gave the first purely analytic proof of both the fundamental theorem of algebra and the intermediate value theorem, and early consideration of sets(collections of objects defined by a common property).

Joseph Fourier

son of a tailor in Auxere

He received his education through the Benedictine Order

Best known for his book Théorieanalytique de la chaleur

Contributions of Fourier

Fourier Series An important advance in mathematical

analysis

Periodic functions that can be expressed as the sum of an infinite series of sines and cosines

It is a powerful tool in pure and applied mathematics.

Fourier series are used in solving differential equations that arise in the study of heat flow and vibration.

The Fourier Series

Johann Peter Gustav Lejeune Dirichlet

Johann Peter Gustav Lejeune Dirichlet

In 1837, he suggested a very broad definition of function,

Dirichlet Function When x is rational, let y = c and when x is

irrational, let y = d ≠ c

He gave the first rigorous proof of the convergence of Fourier Series for a function subject to certain restrictions.

In pure mathematics, he is well – known for his application of analysis to the theory of numbers.

Johann Peter Gustav Lejeune Dirichlet

In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes this form

Bernhard Riemann

Bernhard Riemann

He arrived at deep theorems relating number theory and classical analysis.

Euler had noted connections between prime-number theory and the series

Bernhard Riemann

The Riemann zeta function ζ(s) is a function of a complex variable s = σ + it. The following infinite series converges for all complex numbers s with real part greater than 1, and defines ζ(s) in this case:

Bernhard Riemann

In 1859, Riemann conjectured that all the imaginary zeros of the zeta function have their real part

Contributions

refinement of the definition of the integral by the definition of the Riemann integral

emphasis on the Cauchy – Riemann differential equations

Riemann Surfaces – are ingenious scheme for uniformizing a function

Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.