全学共通科目講義(1回生~4回生対象)
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現代の数学と数理解析 |
―― 基礎概念とその諸科学への広がり |
数学・数理解析の研究の面白さ・深さを、 感性豊かな学生諸君に味わってもらうことを意図して講義し、 原則として予備知識は仮定しない。 | |
日時: | 2016年6月10日(金) 16:30-18:00 |
場所: | 数理解析研究所 420号室 |
講師: | Helmke, Stefan 助教 |
題目: |
A Brief History of Implicit Functions
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要約: |
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)
depending
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. References: Section 2 of the first reference contains a short history of the Implicit Function Theorem, mostly accessible also for beginners. The second reference, Section (III.8.3), contains a nice account on the Puiseux Expansion.
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"http://www.kurims.kyoto-u.ac.jp/ja/special-02.html" |