Theorems in analysis | Inverse functions

Lagrange reversion theorem

In mathematics, the Lagrange reversion theorem gives series or formal power series expansions of certain implicitly defined functions; indeed, of compositions with such functions. Let v be a function of x and y in terms of another function f such that Then for any function g, for small enough y: If g is the identity, this becomes In which case the equation can be derived using perturbation theory. In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation for v mentioned above. However, his solution used cumbersome series expansions of logarithms. In 1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, which was based on relations between partial derivatives with respect to the variable x and the parameter y. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration. Lagrange's reversion theorem is used to obtain numerical solutions to Kepler's equation. (Wikipedia).

Video thumbnail

10 Adjoint state method

We show the connection between the method of adjoints in optimal control to the implicit function theorem ansatz. We relate the costate or adjoint state variable to Lagrange multipliers.

From playlist There and Back Again: A Tale of Slopes and Expectations (NeurIPS-2020 Tutorial)

Video thumbnail

The Cartan Geometry of the Rotating Kepler Problem - Otto van Koert

Otto van Koert Seoul National University January 21, 2011 GEOMETRY/DYNAMICAL SYSTEMS In this talk we shall discuss the Cartan geometry of the rotating Kepler problem. The rotating Kepler problem appears as the limit of the restricted planar three-body body when one of the masses goes to z

From playlist Mathematics

Video thumbnail

Simplifying an expression using the rules of exponents

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

Rules of exponents to help us simplify a rational expression

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

How to simplify an expression using division property of exponents

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

Ivan Guo: Stochastic Optimal Transport in Financial Mathematics

Abstract: In recent years, the field of optimal transport has attracted the attention of many high-profile mathematicians with a wide range of applications. In this talk we will discuss some of its recent applications in financial mathematics, particularly on the problems of model calibra

From playlist SMRI Seminars

Video thumbnail

Applying the quotient rule to simplify a rational expression

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

Two-dimensional stochastic interface growth – Fabio Toninelli – ICM2018

Mathematical Physics Invited Lecture 11.5 Two-dimensional stochastic interface growth Fabio Toninelli Abstract: Stochastic interface dynamics serve as mathematical models for diverse time-dependent physical phenomena: the evolution of boundaries between thermodynamic phases, crystal grow

From playlist Mathematical Physics

Video thumbnail

GT3. Cosets and Lagrange's Theorem

Abstract Algebra: Let G be a group with subgroup H. We define an equivalence relation on G that partitions G into left cosets. We use this partition to prove Lagrange's Theorem and its corollary. U.Reddit course materials available at http://ureddit.com/class/23794/intro-to-group-theor

From playlist Abstract Algebra

Video thumbnail

Lecture 16: Fundamental Welfare Theorems

MIT 14.04 Intermediate Microeconomic Theory, Fall 2020 Instructor: Prof. Robert Townsend View the complete course: https://ocw.mit.edu/courses/14-04-intermediate-microeconomic-theory-fall-2020/ YouTube Playlist: https://www.youtube.com/watch?v=XSTSfCs74bg&list=PLUl4u3cNGP63wnrKge9vllow3Y2

From playlist MIT 14.04 Intermediate Microeconomic Theory, Fall 2020

Video thumbnail

Using the rules of exponents to simplify an expression

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

How do negative exponents simplify an expression

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

Colloquium MathAlp 2016 - Emmanuel Trélat

Théorie du contrôle optimal et applications aux missions spatiales La problématique du contrôle optimal est de guider l'évolution en temps d'un système donné vers une configuration finale souhaitée, tout en minimisant un certain critère. Le point saillant de cette théorie, qui généralise

From playlist Colloquiums MathAlp

Video thumbnail

Rewriting an expression by simplifying using rules of exponents

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

Learn how to rewrite an exponent in the numerator when it has a negative power

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Video thumbnail

Quenched large deviations for random motions in degenerate random media by Chiranjib Mukherjeer

Large deviation theory in statistical physics: Recent advances and future challenges DATE: 14 August 2017 to 13 October 2017 VENUE: Madhava Lecture Hall, ICTS, Bengaluru Large deviation theory made its way into statistical physics as a mathematical framework for studying equilibrium syst

From playlist Large deviation theory in statistical physics: Recent advances and future challenges

Video thumbnail

What is the product of powers of exponents

👉 Learn about the rules of exponents. An exponent is a number which a number is raised to, to produce a power. It is the number of times which a number will multiply itself in a power. There are several rules used in evaluating exponents. Some of the rules includes: the product rule, which

From playlist Simplify Using the Rules of Exponents

Video thumbnail

Non-Hermiticity: A New Paradigm for Model Building in Particle Physics by Peter Millington

PROGRAM NON-HERMITIAN PHYSICS (ONLINE) ORGANIZERS: Manas Kulkarni (ICTS, India) and Bhabani Prasad Mandal (Banaras Hindu University, India) DATE: 22 March 2021 to 26 March 2021 VENUE: Online Non-Hermitian Systems / Open Quantum Systems are not only of fundamental interest in physics a

From playlist Non-Hermitian Physics (ONLINE)

Video thumbnail

History of Mathematics - Complex Analysis Part 2: functions of a complex variable. 3rd Yr Lecture

Complex numbers pervade modern mathematics, but have not always been well understood. They first emerged in the sixteenth century from the study of polynomial equations, and were quickly recognised as useful – if slightly weird – mathematical tools. In these lectures (this is the second

From playlist Oxford Mathematics 3rd Year Student Lectures

Video thumbnail

Simplifying expressions using the rules of exponents, quotient property

👉 Learn how to simplify expressions using the quotient rule and the negative exponent rule of exponents. The quotient rule of exponents states that the quotient of powers with a common base is equivalent to the power with the common base and an exponent which is the difference of the expon

From playlist Simplify Using the Rules of Exponents

Related pages

Series (mathematics) | Perturbation theory | Charles Hermite | Formal power series | Mathematics | Pierre-Simon Laplace