Editing Augustin-Louis Cauchy (section)

==Work==

===Early work===
The genius of Cauchy was illustrated in his simple solution of the [[problem of Apollonius]]—describing a [[circle]] touching three given circles—which he discovered in 1805, his generalization of [[Euler characteristic|Euler's formula]] on [[polyhedra]] in 1811, and in several other elegant problems. More important is his memoir on [[wave]] propagation, which obtained the Grand Prix of the French Academy of Sciences in 1816. Cauchy's writings covered notable topics. In the theory of series he developed the notion of [[limit of a sequence|convergence]] and discovered many of the basic formulas for [[q-series]]. In the theory of numbers and complex quantities, he was the first to define [[complex number]]s as pairs of real numbers. He also wrote on the theory of groups and substitutions, the theory of functions, differential equations and determinants.{{sfn|Chisholm|1911}}

===Wave theory, mechanics, elasticity===
In the theory of light he worked on [[Augustin-Jean Fresnel|Fresnel's]] wave theory and on the [[dispersion (optics)|dispersion]] and [[polarization (waves)|polarization]] of light. <!-- In [[optics]], he developed the wave theory, and his name is associated with the simple dispersion formula. Exactly which formula? --> He also contributed research in [[mechanics]], substituting the notion of the continuity of geometrical displacements for the principle of the continuity of matter.<ref>{{cite book |last=Kurrer |author-link=Karl-Eugen Kurrer |first=K.-E. |date=2018 |title=The History of the Theory of Structures. Searching for Equilibrium |location=Berlin |publisher=[[John Wiley & Sons|Wiley]]|pages=978–979 |isbn=978-3-433-03229-9}}</ref> He wrote on the equilibrium of rods and elastic membranes and on waves in elastic media. He introduced a 3 × 3 symmetric [[matrix (mathematics)|matrix]] of numbers that is now known as the [[Cauchy stress tensor]].{{sfn|Cauchy|1827|p=42|loc="''De la pression ou tension dans un corps solide''" [On pressure or tension in a solid body]}} In [[Elasticity (physics)|elasticity]], he originated the theory of [[stress (physics)|stress]], and his results are nearly as valuable as those of [[Siméon Poisson]].{{sfn|Chisholm|1911}}

===Number theory===
Other significant contributions include being the first to prove the [[Fermat polygonal number theorem]].

===Complex functions===
Cauchy is most famous for his single-handed development of [[complex function theory]]. The first pivotal theorem proved by Cauchy, now known as ''[[Cauchy's integral theorem]]'', was the following:

:<math>
 \oint_C f(z)dz = 0,
</math>

where ''f''(''z'') is a complex-valued function [[holomorphic function|holomorphic]] on and within the non-self-intersecting closed curve ''C'' (contour) lying in the [[complex plane]]. The ''contour integral'' is taken along the contour ''C''. The rudiments of this theorem can already be found in a paper that the 24-year-old Cauchy presented to the Académie des Sciences (then still called "First Class of the Institute") on August 11, 1814. In full form the theorem was given in 1825.{{sfn|Cauchy|1825}}

In 1826 Cauchy gave a formal definition of a [[residue (mathematics)|residue]] of a function.{{sfn|Cauchy|1826|p=11|loc="''Sur un nouveau genre de calcul analogue au calcul infinitésimal''" [On a new type of calculus analogous to the infinitesimal calculus]}} This concept concerns functions that have [[pole (complex analysis)|pole]]s—isolated singularities, i.e., points where a function goes to positive or negative infinity. If the complex-valued function ''f''(''z'') can be expanded in the [[neighborhood]]  of a singularity ''a'' as

:<math>
f(z) = \varphi(z) + \frac{B_1}{z-a} + \frac{B_2}{(z-a)^2} + \cdots + \frac{B_n}{(z-a)^n},\quad
B_i, z,a \in \mathbb{C},
</math>

where φ(''z'') is analytic (i.e., well-behaved without singularities), then ''f'' is said to have a pole of order ''n'' in the point ''a''. If ''n'' = 1, the pole is called simple.
The coefficient ''B''<sub>1</sub> is called by Cauchy the residue of function ''f'' at ''a''. If ''f'' is non-singular at ''a'' then the residue of ''f'' is zero at ''a''. Clearly, the residue is in the case of a simple pole equal to
:<math>
\underset{z=a}{\mathrm{Res}} f(z) = \lim_{z \rightarrow a} (z-a) f(z),
</math>
where we replaced ''B''<sub>1</sub> by the modern notation of the residue.

In 1831, while in Turin, Cauchy submitted two papers to the Academy of Sciences of Turin. In the first{{sfn|Cauchy|1831}} he proposed the formula now known as [[Cauchy's integral formula]],
:<math>
f(a) = \frac{1}{2\pi i} \oint_C \frac{f(z)}{z-a} dz,
</math>
where ''f''(''z'') is analytic on ''C'' and within the region bounded by the contour ''C'' and the complex number ''a'' is somewhere in this region. The contour integral is taken counter-clockwise. Clearly, the integrand has a simple pole at ''z'' = ''a''. In the second paper<ref>Cauchy, ''Mémoire sur les rapports qui existent entre le calcul des Résidus et le calcul des Limites, et sur les avantages qu'offrent ces deux calculs dans la résolution des équations algébriques ou transcendantes'' (Memorandum on the connections that exist between the residue calculus and the limit calculus, and on the advantages that these two calculi offer in solving algebraic and transcendental equations], presented to the Academy of Sciences of Turin, November 27, 1831.</ref> he presented the [[residue theorem]],
:<math>
 \frac{1}{2\pi i} \oint_C f(z) dz = \sum_{k=1}^n \underset{z=a_k}{\mathrm{Res}} f(z),
</math>
where the sum is over all the ''n'' poles of ''f''(''z'') on and within the contour ''C''. These results of Cauchy's still form the core of complex function theory as it is taught today to physicists and electrical engineers. For quite some time, contemporaries of Cauchy ignored his theory, believing it to be too complicated. Only in the 1840s the theory started to get response, with [[Pierre Alphonse Laurent]] being the first mathematician besides Cauchy to make a substantial contribution (his work on what are now known as [[Laurent series]], published in 1843).

===''Cours d'analyse''===
{{Main|Cours d'analyse}}
[[Image:Cauchy.jpg|left|thumb|The title page of a textbook by Cauchy.]] In his book ''Cours d'analyse'' Cauchy stressed the importance of rigor in analysis. ''Rigor'' in this case meant the rejection of the principle of ''[[Generality of algebra]]'' (of earlier authors such as Euler and Lagrange) and its replacement by geometry and [[infinitesimal]]s.{{sfn|Borovik|Katz|2012|pp=245–276}} Judith Grabiner wrote Cauchy was "the man who taught rigorous analysis to all of Europe".{{sfn|Grabiner|1981}}  The book is frequently noted as being the first place that inequalities, and <math>\delta-\varepsilon</math> arguments were introduced into calculus.  Here Cauchy defined continuity as follows: ''The function f(x) is continuous with respect to x between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself.''

M. Barany claims that the École mandated the inclusion of infinitesimal methods against Cauchy's better judgement.{{sfn|Barany|2011}}  Gilain notes that when the portion of the curriculum devoted to ''Analyse Algébrique'' was reduced in 1825, Cauchy insisted on placing the topic of continuous functions (and therefore also infinitesimals) at the beginning of the Differential Calculus.{{sfn|Gilain|1989}} Laugwitz (1989) and Benis-Sinaceur (1973) point out that Cauchy continued to use infinitesimals in his own research as late as 1853.

Cauchy gave an explicit definition of an infinitesimal in terms of a sequence tending to zero.  There has been a vast body of literature written about Cauchy's notion of "infinitesimally small quantities", arguing that they lead from everything from the usual "epsilontic" definitions or to the notions of [[non-standard analysis]]. The consensus is that Cauchy omitted or left implicit the important ideas to make clear the precise meaning of the infinitely small quantities he used.{{sfn|Barany|2013}}

===Taylor's theorem===
He was the first to prove [[Taylor's theorem]] rigorously, establishing his well-known form of the remainder.{{sfn|Chisholm|1911}} He wrote a textbook{{sfn|Cauchy|1821}} (see the illustration) for his students at the École Polytechnique in which he developed the basic theorems of mathematical analysis as rigorously as possible. In this book he gave the necessary and sufficient condition for the existence of a [[limit of a function|limit]] in the form that is still taught.<!-- which is epsilon delta? --> Also Cauchy's well-known test for [[absolute convergence]] stems from this book: [[Cauchy condensation test]]. In 1829 he defined for the first time a complex function of a complex variable in another textbook.{{sfn|Cauchy|1829}} In spite of these, Cauchy's own research papers often used intuitive, not rigorous, methods;{{sfn|Kline|1982|p=176}} thus one of his theorems was exposed to a "counter-example" by [[Niels Henrik Abel|Abel]], later fixed by the introduction of the notion of [[uniform continuity]].

===Argument principle, stability===
In a paper published in 1855, two years before Cauchy's death, he discussed some theorems, one of which is similar to the "[[Argument principle|Principle of the argument]]" in many modern textbooks on complex analysis. In modern control theory textbooks, the [[Cauchy argument principle]] is quite frequently used to derive the [[Nyquist stability criterion]], which can be used to predict the stability of negative [[feedback amplifier]] and negative [[feedback]] control systems. Thus Cauchy's work has a strong impact on both pure mathematics and practical engineering.