Mathematics and Its History pp 313-334 | Cite as

# Complex Numbers and Functions

## Preview

The insight into algebraic curves afforded by complex coordinates—that a complex curve is topologically a surface—has important repercussions for functions defined as integrals of algebraic functions, such as the logarithm, exponential, and elliptic functions. The complex logarithm turns out to be “many-valued,” due to the different paths of integration in the complex plane between the same endpoints. It follows that its inverse function, the exponential function, is *periodic*. In fact, the complex exponential function is a fusion of the real exponential function with the sine and cosine: *e* ^{ x+iy } = *e* ^{ x }(cos *y* + *i* sin *y*). The double periodicity of elliptic functions also becomes clear from the complex viewpoint. The integrals that define them are taken over paths on a *torus* surface, on which there are two independent closed paths. The two-dimensional nature of complex numbers imposes interesting and useful constraints on the nature of *differentiable* complex functions. Such functions define *conformal* (angle-preserving) maps between surfaces. Also, their real and imaginary parts satisfy equations, called the *Cauchy–Riemann* equations, that govern fluid flow. So complex functions can be used to study the motion of fluids. Finally, the Cauchy–Riemann equations imply *Cauchy’s theorem*. This fundamental theorem guarantees that differentiable complex functions have many good features, such as power series expansions.

## Keywords

Conformal Mapping Complex Function Elliptic Curf Elliptic Function Algebraic Curf## Preview

Unable to display preview. Download preview PDF.

## References

- Clebsch, A. (1864). Über einen Satz von Steiner und einige Punkte der Theorie der Curven dritter Ordnung.
*J. reine und angew. Math.**63*, 94–121.zbMATHGoogle Scholar - Dedron, P. and J. Itard (1973).
*Mathematics and Mathematicians, Vol. 1*. Milton Keynes: Open University Press.Google Scholar - Eisenstein, G. (1847). Beiträge zur Theorie der elliptische Functionen.
*J. reine und angew. Math.**35*, 137–274.zbMATHGoogle Scholar - Goursat, E. (1900). Sur la définition générale des fonctions analytiques, d’après Cauchy.
*Trans. Amer. Math. Soc.**1*, 14–16.zbMATHMathSciNetGoogle Scholar - McKean, H. and V. Moll (1997).
*Elliptic Curves*. Cambridge: Cambridge University Press.zbMATHGoogle Scholar - Needham, T. (1997).
*Visual Complex Analysis*. Oxford: Clarendon Press.zbMATHGoogle Scholar - Puiseux, V.-A. (1850). Recherches sur les fonctions algébriques.
*J. Math.**15*, 365–480.Google Scholar - Salmon, G. (1851). Théorèmes sur les courbes de troisième degré.
*J. reine und angew. Math.**42*, 274–276.zbMATHGoogle Scholar