# Functional equations
---
1. Integer functional equations 
2. Rational functional equations 
3. Real functional equations
4. Multivariate functional equations


\chapter{Functional equations}

Functional equations are equations in terms of an unknown function, where we would ideally like to find such a satisfying function.
We can consider functions with various types of domains and codomains with algebraic structure, and when applicable, we can assume certain analytic properties of satisfying functions (continuous, differentiable, analytic, etc.)

Notably, differential equations are a special type of functional equation, the study of such functional equations is not documented here as it is in another book of mine.

Algebraic properties
Topological properties
Ordering properties
Analytic properties


Readers are expected to be familiar with basic types of functions.

even functions
odd functions
periodic functions
idempotent functions
Recurrence relations


Certain function properties incur various branches of mathematics to become useful to different functional equations. Indeed, the theory of functional equations is a ubiquitous in mathematics.


			<h3>Functional equation</h3>
			<ul>
				<li>Difference equation</li>
				<li>Differential equation</li>
				<li>Integral equation</li>
				<li>Integral equation</li>
			</ul>


			<h3>Homogeneous</h3>
			<h3>Inhomogeneous</h3>



\begin{definition}[Schröder's equation]
			<p>\( \forall x, f(g(x)) = kf(x) \)</p>
			<ul>
				<li>\(g\) is a given function</li>
			</ul>






			<h2>Functional Properties</h2>


			<h4>Even</h4>
			\[ f \text{ is even } \iff f(-x)=f(x) \]
			<h4>Odd</h4>
		\[ f \text{ is odd } \iff f(-x)=-f(x) \]
			<h4>Periodic</h4>
			\[ f \text{ is } $k$\text{-periodic } \iff \exists k [ \forall x \in \mathbf{dom}(f)[ f(x)=f(x+k) ]]\]
			<h4>Polynomial function</h4>
			\[ f(x) = \sum^{n}_{k=0} c_i x^k \]


			<h4>Involuntory function</h4>

			<h4>Additive</h4>
			<h4>Sub-additive</h4>
			<h4>Super-additive</h4>
			<h4>\(\sigma\)-additive</h4>




\chapter{Difference equations}

\section{Elementary difference equations}
\[f : \mathbb{N} \to \mathbb{C}\]
\[f(n) = k f(n-1)\]
\[f(n) = k^n f(0) \]


\section{Transforming difference equations into eigenequations}