|講師：||Helmke, Stefan 助教|
A Brief History of Implicit Functions
An implicit function is a function that is implicitly defined
as the solution of an equation with parameters. For example, if
F (x, y) is a function of two variables,
we may consider the first
variable x as a parameter and try to find a function h (x)
only on this parameter x, such that the equation
F (x, h (x))=0 is
fulfilled, for all x. In other words, we try to solve the equation
F (x, y) = 0 in the form y = h (x).
The Implicit Function Theorem
gives sufficient conditions for the equation F, which guarantee
the existence of a solution h locally. The modern version of this
theorem was first proved by Ulisse Dini (1845--1918) in the 1870's.
But already Isaac Newton (1642--1727) considered this problem in
the 17th century and gave for h an elegant fractional power series
solution, the so-called Puiseux Expansion (named after the
French mathematician Victor Alexandre Puiseux (1820--1883).)
In this lecture I will explain Newton's solution to the problem, the historic developments with many examples, which led to Dini's theorem and some further developments.