Implicit function on functions composed of itself

Thread starter chy1013m1
Start date Jan 13, 2007
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Function Functions Implicit

In summary, an implicit function composed of itself is a recursive function where the output depends on its previous output. This type of function differs from traditional ones as it involves self-reference, making it more complex to analyze and solve. However, implicit functions on functions composed of itself have significant applications in fields such as computer science, economics, and physics, as they can model feedback loops and self-referential behavior. Studying these functions presents challenges such as potential for infinite loops and non-convergent solutions, as well as difficulties in analyzing their behavior and finding solutions. In real-world scenarios, implicit functions on functions composed of itself are used in the study of chaotic systems, population dynamics, neural networks, and decision-making processes, among others.

Jan 13, 2007

chy1013m1

Suppose F(x, y) is C1. F(0, 0) = 0. What conditions on F will guarantee that the equation F(F(x, y), y) = 0 can be solved for y as a C1 function of x near (0, 0) ?

would it simply be dF/dy not equal 0 ?

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Jan 13, 2007

Tom1992

Use the implicit function theorem. Seems you are on the right track, but F is a composition of F and y, so you would have to use the chain rule to right out dF/dy properly.

FAQ: Implicit function on functions composed of itself

What is an implicit function composed of itself?

An implicit function composed of itself is a function where the dependent variable is also used as the independent variable. This means that the function is recursive, and its output depends on its previous output.

How do implicit functions on functions composed of itself differ from traditional functions?

Implicit functions on functions composed of itself differ from traditional functions in that they involve self-reference, leading to a recursive structure. This can make them more complex to analyze and solve compared to traditional functions.

What is the significance of implicit functions on functions composed of itself?

Implicit functions on functions composed of itself have applications in various fields, such as computer science, economics, and physics. They can model complex systems and processes that involve feedback loops and self-referential behavior.

What are the challenges in studying implicit functions on functions composed of itself?

One of the main challenges in studying implicit functions on functions composed of itself is the potential for infinite loops and non-convergent solutions. Additionally, analyzing their behavior and finding closed-form solutions can be difficult and require advanced mathematical techniques.

How are implicit functions on functions composed of itself used in real-world scenarios?

Implicit functions on functions composed of itself are used in various real-world scenarios, such as in the study of chaotic systems, population dynamics, and neural networks. They can also be used to model decision-making processes and feedback mechanisms in economics and social sciences.