This was a project for a history of science class I took. I was rather happy with the final product, so why not share it!
Category Archives: Science and Math
A Brief History of Complex Analysis in the 19th Century
Until the turn of the 19th century, the majority of mathematical study since Newton’s Principia was grounded in real analysis with applications to physical phenomenon. The Norwegian mathematician Niels Henrik Abel (“Niels Henrik Abel,” n.d.) was in Paris in 1826 when he presents this observation in a letter to his teacher, “He is moreover the only one today working in pure mathematics. Poisson, Fourier, Ampere, etc. etc. occupy themselves with nothing other than magnetism and other physical matters.” (Bottazzini 1986, 85) The person Abel is referring to is the French mathematician Augustin-Louis Cauchy. Cauchy was born in 1780 in Paris, France, graduated from the Ecole Polytechnique in 1807 and after a brief stint as an engineer returned to the Ecole Polytechnique in 1815 to teach (Bottazzini 1986, 101).
In 1821 the French mathematician Augustin-Louis Cauchy publish a series of lectures titled Cours d’analyse(Bottazzini 1986, 101). The Cours d’analyse was a rigorous text that quickly became the manifesto of analysis in its time. Cauchy’s most significant contribution to mathematics was his work in complex analysis, to which the later half of his Cours d’analyse is dedicated. Cauchy introduces the a+ib form of the complex number (where i represents the square root of -1), then proceeds to set out the framework that defines the properties and operations on such quantities. Cauchy then reinterprets the concepts of real variable analysis using complex variables to provide demonstrations regarding continuity of functions, series and convergence. The Cours d’analyse stopped with the treatment of the differential calculus and did not address the problem of complex integration.
Cauchy’s first work on complex integration appeared in an 1814 paper on definite integrals (improper real integrals) that was presented to the Institute but not published until 1827 (Bottazzini 1986, 132). In this paper Cauchy describes the method passing from the real to the imaginary realm where one can then calculate an improper integral with ease. This is the first hint of Cauchy’s later famous integral formula and Cauchy-Riemann equations. In 1823 Cauchy published the Resume which picked up where his Cours d’analyse left off, addressing complex integration and introducing his integration formula (Bottazzini 1986, 142). By the end of the 1820’s Cauchy had established the field of complex analysis, all of which was based in pure mathematical theory.
Cauchy’s approach and related work on complex analysis, though rigorous and thorough, focused very little on the subject of complex geometry. In the beginning of the 19th century there had been some work done regarding the geometrical representation of complex numbers in the plane, most notably an 1806 paper by the Swiss mathematician Jean Robert Argand (Kreyszig 2011, 611). It wasn’t until 1831 when Gauss finally published his work from almost 30 years prior that the geometrical theory of complex numbers gained general acceptability (Bottazzini 1986, 166). Cauchy, reluctant at first, eventually redefined much of his earlier works in the later half of the 1840’s to incorporate the new geometrical representations of the complex numbers.
The next big development in complex analysis came from Gauss’ student, Bernhard Riemann. Riemann was born in 1826 in the Kingdom of Hanover and studied at Gottingen where he presented his dissertation in 1851 (Laugwitz 1999, 1). Riemann’s dissertation, Foundations for a general theory of functions of a complex variable, takes a completely new, geometric approach to complex analysis and introduces what are called Riemann surfaces (Laugwitz 1999, 96). The Riemann surface is a new and novel idea in mathematics as noted by the 20th century Finnish mathematician Lars Ahlfors:
The reader is led to believe that this is a commonplace convention, but there is no record of anyone having used a similar device before. As used by Riemann it is a skillful fusion of two distinct and equally import ideas: 1) A purely topological notion of covering surface, necessary to clarify the concept of mapping in the sense of multiple correspondence; 2) An abstract conception of the space of the variable; with a local structure defined by a uniformizing parameter. (1953, 4)
Riemann further introduces the definition of holomorphic functions as complex single-valued functions on Riemann surfaces which satisfy the Cauchy-Riemann equations, which gives way to the representing domains through conformal mappings (Laugwitz 1999, 99). At the end of his dissertation Riemann introduces his mapping theorem:
Two given simply connected plane surfaces can always be related to each other in such a way that each point of one surface corresponds to a point of the other, varying continuously with that point, with the corresponding smallest parts similar.
Riemann 2004, 37
as the solution to the problem of conformally mapping two Riemann surfaces.
The geometric genius of Riemann is further exploited with his introduction of what is now referred to as a topological space, but described by Riemann as a manifold of n real coordinates (Laugwitz 1999, 231). The work done in pure differential geometry by Riemann was further developed throughout the 19th and 20th centuries into modern topology.
The developments made in complex analysis soon found their way to the world of physics. By the mid 19th century potential theory was well established in the areas of gravitation and electrostatics, governed by Laplace’s equation whose solutions are harmonic functions. Ahlfors sums up the relationship between potential theory and complex analysis when he said, Riemann “virtually puts equality signs between two-dimensional potential theory and complex function theory” (1953, 4). Later applications in physics include solutions to Schrödinger’s equation, fluid flow, string theory and Einstein’s General Theory of Relativity.
References
- Ahlfors, L. V. (1953). Development of the Theory of Conformal Mapping and Riemann Surfaces through a Century. Ann. of Math. Stud. 30, pp. 3-13.
- Bottazzini, Umberto. (1986). The Higher Calculus: A History of Real and Complex Analysis form Euler to Weierstrass. New York: Springer-Verlag.
- Kreyszig, Erwin. (2011). Advanced Engineering Mathematics. 10th ed. Hoboken, NJ: John Wiley & Sons, Inc.
- Laugwitz, Detlef. (1999). Bernhard Riemann 1816-1866, Turning Points in the Conception of Mathematics. (Abe Shenitzer, Trans.). Boston, MA: Birkhauser.
- Niels Henrik Abel. (2013, October 12). In Wikipedia, The Free Encyclopedia. Retrieved 01:06, October 12, 2013, from http://en.wikipedia.org/w/index.php? title=Niels_Henrik_Abel&oldid=577314187
- Riemann, Bernhard. (2004). Collected Papers. (Roger Baker, Charles Christenson and Henry Order, Trans.). Heber City, UT: Kendrick Press.