## History

Euler wrote six books on calculus between 1748 and 1770. All six were written in Latin, the lingua franca of scholars of the 18th century. There were two volumes entitled \textsl{Introduction to analysis of the infinite}, one volume on \textsl{Methods of differential calculus}, and three volumes on \textsl{Methods of the integral calculus}. Euler made functions the central idea in his books. This was a radical move, since both Newton and Leibniz in their development of the calculus had dealt with curves. The idea of a function was not fully developed in these books, since Euler restricted a function to mean a formula. However, he did consider functions of a complex variable as well as functions of a real variable. He also considered divergent series as well as convergent series.

Euler divided the set of functions into two basic classes: algebraic and transcendental. Algebraic functions are formed from variables and constants by addition, subtraction, multiplication, division, raising to an integer power, and extraction of roots. For Euler, the transcendental functions were limited to trigonometric, exponential and logarithmic functions.

Euler made extensive use of infinite series, infinite products and infinite continued fractions. As an example of the depth of his thinking, he was the first mathematician to seriously investigate the Riemann zeta function, defined by

$\zeta(s) = \sum_{n=1}^\infty \dfrac{1}{n^s}.$

For example, Euler showed that

$\zeta(2) = \sum_{n=1}^\infty \dfrac{1}{n^2} = \dfrac{\pi^2}{6}.$

In the 19th century, the centrality of the complex numbers to mathematics, and in particular to both algebra and analysis (calculus), was realised. Functions of a complex variable were introduced, and it was shown (essentially) that any function given by a formula such as $$z^2$$ or $$\sin z$$, where $$z=x+iy$$, could be both integrated and differentiated. This material is typically taught in second-year university courses. (Reference: R. V. Churchill and J. W. Brown, Complex variables and applications, 5th edition, McGraw-Hill, 1990.)

In the 20th century, the idea of a function was generalised considerably further, leading to two branches of modern analysis called harmonic analysis and functional analysis.

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