Definition:Function/Historical Note
Historical Note on Function
The term function, as used in the modern sense, was first used by Gottfried Wilhelm von Leibniz in $1694$.
The notation $\map f x$ itself appears to have originated with Leonhard Paul Euler.
He used it in two particular contexts:
- particular conventional examples like trigonometric function and powers and the like
- $\map y x$ for an arbitrary curve in the plane.
Up until the time of Joseph Fourier, it was accepted that a function was limited to various classes of expression: a polynomial, a finite combination of elementary functions, a power series or a trigonometric series.
Fourier made the claim that a function of arbitrary shape could be represented by a trigonometric series.
It was not until Johann Peter Gustav Lejeune Dirichlet in $1837$ that the modern definition of function was formulated:
- If in any way a definite value of $y$ is determined corresponding to each value of $x$ in a given interval, then $y$ is called a function of $x$.
The concept of a mapping between arbitrary sets which are not necessarily the real or complex numbers arose in the late $19$th century.
Sources
- 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
- 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 1$: Introduction
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.19$: Leibniz ($\text {1646}$ – $\text {1716}$)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.24$: Fourier ($\text {1768}$ – $\text {1830}$)
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.28$: Dirichlet ($\text {1805}$ – $\text {1859}$)