# Cauchys Contributions in Mathematics

Document Type:Thesis

Subject Area:Mathematics

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Cauchy contributed immensely in different fields including the following; mathematics, physics, mechanics, and celestial mechanics. However, this paper focuses on his contributions to mathematics. Calculus In Calculus Cauchy developed various calculus theorems and published them in a book referred to as calculus 1 and 2, which was later adopted and used in Cours d’analyse that was put together in 1821 (Cauchy, Kappel & Schappacher, 1981). Various mathematicians believed his analysis provided the breakthrough to higher concepts in calculus. The approach utilized therein included ideas drawn from geometry and algebra. Cauchy provided an elaborate definition of the upper and lower limits. He is believed to be the first person to show the convergence of (1 + 1/n) n. Equally, he is credited with being the first person to employ the limit sign in doing calculations.

Cauchy applied rearrangement, addition, and multiplication in exploiting various dynamics of convergence of series. He was cautious in his explanation of convergence of double series so that to avoid possibilities of leaving critical ideas outside (Cauchy & Cates, 2012). Cauchy emphasized on the areas captured by differential quotient and provided a description that adequately covered the idea of the definite integral. He explored improper integrals, where he also explained his principal value that captured an integral that was made up of a singular integrand (Grabiner, 1994). He exhaustively brought into view discontinuous factors and the concept that expounded more on the Fourier transform. Besides, he developed the Jacobian principle, which was, however, limited to two and three perspectives. Furthermore, he proved the fundamental theorem of algebra that utilizes the tool of decreasing the absolute digit found in an analytic function provided it does not disappear.

Utilization of geometrical principles to interpret complex numbers was popularized by Gauss (Cauchy, Bradley & Sandifer, 2009). There are no records; however, showing if earlier scholars who viewed complex numbers as real functions were conversant with the method that used geometrical techniques to prove the validity of complex functions. Even though Gauss proved the fundamental theorem of algebra in the real platform, he still captured some ideas from complex number theorem in his explanations. Rash ideas of Euler are probably the most exhaustive frameworks that were used to study complex functions during Cauchy’s era. Besides, Laplace concept of changing the course of real integration in the complex forum was utilized to get new approaches for definite integrals, which was a procedure that existed without any form of proof whatsoever.

Despite his extensive involvement in the field, Cauchy was unable to comfortably dissect his concept through a complex domain. He perceived the complex functions as two variables that existed in pairs, which the principle that the Cauchy-Riemann differential equations employ. Instead of justifying the complex approach, Cauchy decided to bypass it. He further explained his views on complex numbers by criticizing the opinion advanced by Legendre. Error Theory Cauchy’s contribution in error theory is insurmountable. In 1837, despite its inefficiency compared to the least square method, Cauchy tried to peddle the ideas in pre-Laplacian technique (Cauchy, Kappel & Schappacher, 1981). He argued by showing that maximal errors can be reduced in the worst possible circumstances. Although it is not sufficiently clear, it is a concept that is vague enough to prove the dynamics in the older approaches.

In his new method, Cauchy found himself dealing with a unique problem, where he was required to fit his set of observational data to polynomials. Such as the following; U = ax + by + cz… (Fraser, 1995) Here, the number of values exclusively depended on the goodness of fit that was realized when the calculations were done. However, this is considered to be an unsustainable postulate because the resultant ki is made to be dependent on the kind of η selected. However, Cauchy wanted to advance an idea that ensured the kind of f chosen is not in any way dependent on η. His observation seemed dented because f is not considered as a tool of the observer rather an entity of nature; however, it gives a good outcome.

It is notable that the only f that complies with this criteria is the one applied in a Fourier transform ϕ where; Φ (ξ) = exp (−αξN), Where α and N are constants (Cauchy, Kappel & Schappacher, 1981). Bienayme noticed that the elements exhibited a paradoxical character where there were unable to be increased by averaging. Through his journal, the phrase “determinate” gained popularity. Although, he later preferred the term “resultant,” this seemed so strange to other scholars. He explores a more abstract technique in handling determinants, just like other popular mathematicians, for example, Grassmann’s. He provides a well-structured and systematic theorem of complex numbers. Equally, he exploited the geometric method using by employing abstract algebraic. Moreover, it demonstrates Cauchy’s theory that exploits finite groups; to all values of p, whereby there is an item of order p that divides the prime p.

Sylow supports the ideas illustrated in this concept. In 1812, Cauchy criticized the facets of Fermat theorem that explores the polygonal of numbers. He argued that each positive integer is a summation of n gonal digits. During that era proofs for n=3, 4 were popular. Second, that distinctiveness has to be demonstrated by making specific initial information instead of insignificant integration constants. The second idea is the most popular and it closely attached to Cauchy issue in incomplete differential equations. This concept is believed to have emerged when Cauchy was conducting his first evaluation of waves in liquids between 1815 and 1816. Cauchy thought that the existing problem had to be pursued in a graphical frame that involved differential equation whose initial limits and data were well-established.

Cauchy was able to navigate through differential equations through the Cauchy –Lipschitz approach of approximating the difference of equations; however, the proof of this method was accepted and published in 1840 (Cauchy, Bradley & Sandifer, 2009). The most consistent tool that was employed in the study of differential equations was the Fourier transform. The framework is evident in all Cauchy’s explorations starting from his research on waves in liquids. His other works more so of 1817 and 1818 proved his persistence in utilizing the concept. Although Fourier was discovered in 1807, it was published in 1826 (Cauchy & Cates, 2012). Therefore, the claim advanced by Cauchy that he formulated the inversion technique independently is agreeable. , & Cates, D. A Guide To Cauchy's Calculus. Walnut Creek, CA: Fairview Academic Press. Cauchy, A.

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