Real analysis in the context of "Functions of a complex variable"

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of a complex variable of complex numbers. It is helpful in many branches of mathematics, including real analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

At first glance, complex analysis is the study of holomorphic functions that are the differentiable functions of a complex variable. By contrast with the real case, a holomorphic functions is always infinitely differentiable and equal to the sum of its Taylor series in some neighborhood of each point of its domain.This makes methods and results of complex analysis significantly different from that of real analysis. In particular, contrarily, with the real case, the domain of every holomorphic function can be uniquely extended to almost the whole complex plane. This implies that the study of real analytic functions needs often the power of complex analysis. This is, in particular, the case in analytic combinatorics.

In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic.

Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of mathematics. In his first theorem, Gödel showed that any consistent system with a computable set of axioms which is capable of expressing arithmetic can never be complete: it is possible to construct a statement that can be shown to be true, but that cannot be derived from the formal rules of the system. In his second theorem, he showed that such a system could not prove its own consistency, so it certainly cannot be used to prove the consistency of anything stronger with certainty. This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of itself, and therefore could not prove everything else.

In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions.

Hans Hahn (/hɑːn/; German: [haːn]; 27 September 1879 – 24 July 1934) was an Austrian mathematician and philosopher who made contributions to functional analysis, topology, set theory, the calculus of variations, real analysis, and order theory. In philosophy he was among the main logical positivists of the Vienna Circle.

Baron Augustin-Louis Cauchy FRS FRSE (UK: /ˈkoʊʃi/ KOH-shee, /ˈkaʊʃi / KOW-shee, US: /koʊˈʃiː / koh-SHEE; French: [oɡystɛ̃ lwi koʃi]; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real analysis), pioneered the field complex analysis, and the study of permutation groups in abstract algebra. Cauchy also contributed to a number of topics in mathematical physics, notably continuum mechanics.

A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated:

In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex-valued functions may be easily reduced to the study of the real-valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real-valued functions will be considered in this article.

The domain of a function of n variables is the subset of ⁠${\displaystyle \mathbb {R} ^{n}}$⁠ for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain a nonempty open subset of ⁠${\displaystyle \mathbb {R} ^{n}}$⁠.

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Göttingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration, or simulated using Monte Carlo integration.