Analysis is the branch of mathematics concerned particularly with the concepts of function and limit. The subject has its origins in the quest to put calculus on a rigorous footing, and it is to this end that concepts like continuous and limit were first defined rigorously by Karl Weierstrass and Augustin-Louis Cauchy. Weierstrass gave the now-familiar "epsilon-delta" definition of a limit and worked to elaborate its basic properties. He demonstrated, for example, that there exist functions which are continuous functions which are not differentiable at any point. The possibility of such pathological functions could not have been imagined by Isaac Newton and others who had worked on calculus with less formal underpinnings.
Since its origins in the calculus, analysis has expanded into a vast subject with applications to every other branch of mathematics. It now includes other familiar topics like Riemann integration and Lebesgue integration, picking up entire fields like measure theory along the way. Analysis also plays an important role in applied mathematics, where it provides the machinery which make methods like Fourier analysis possible, and many deep results about solutions of differential equations may be proved by analytic methods. Besides these well-known subjects, there are numerous other subfields of analysis dealing with more specialized subjects:
- Real analysis, the study of functions of real variables. Real analysis includes most of basic calculus.
- Complex analysis, the study of functions of holomorphic functions of complex variables.
- Functional analysis, the study of spaces of functions, a critical ingredient much of physics.
- Harmonic analysis, dealing with Fourier series and their generalizations.
- Numerical analysis, examining algorithms used for a variety of computations.
In addition to these branches, there are other less familiar branches of analysis, including p-adic and non-standard analysis.
The New American Desk Encyclopedia, Penguin Group, 1989