Editing Functional equation (section)

{{Short description|Equation whose unknown is a function}}
{{distinguish|Functional model}}

In [[mathematics]], a '''functional equation'''
<ref name="rassias">{{cite book | title=Functional Equations and Inequalities | last=Rassias | first=Themistocles M. | year=2000 | publisher=[[Kluwer Academic Publishers]] | location=3300 AA Dordrecht, The Netherlands | isbn=0-7923-6484-8 | page= 335 | url=https://books.google.com/books?id=tFTFBAAAQBAJ&q=%22Introduction+to+the+Theory+of+Functional+Equations+and+Inequalities%22 }}</ref><ref name="rassias4"> {{cite book |title=Functional Equations and Inequalities in Several Variables |last=Czerwik |first=Stephan |year=2002 |publisher=[[World Scientific Publishing Co.]] |location=P O Box 128, Farrer Road, Singapore 912805 |isbn=981-02-4837-7 |page= [https://archive.org/details/functionalequati00czer_083/page/n419 410] |url=https://archive.org/details/functionalequati00czer_083 |url-access=limited }}</ref>{{irrelevant citation|reason=What fact or sentence is being cited here? These citations don't make any sense.|date=March 2022}} is, in the broadest meaning, an [[equation]] in which one or several functions appear as [[unknown (mathematics)|unknown]]s. So, [[differential equation]]s and [[integral equation]]s are functional equations. However, a more restricted meaning is often used, where a ''functional equation'' is an equation that relates several values of the same function. For example, the [[logarithm function]]s are [[logarithm#Characterization by the product formula|essentially characterized]] by the ''logarithmic functional equation'' <math>\log(xy)=\log(x) + \log(y).</math>

If the [[domain of a function|domain]] of the unknown function is supposed to be the [[natural number]]s, the function is generally viewed as a [[sequence (mathematics)|sequence]], and, in this case, a functional equation (in the narrower meaning) is called a [[recurrence relation]]. Thus the term ''functional equation'' is used mainly for [[real function]]s and [[complex function]]s. Moreover a [[smooth function|smoothness condition]] is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the  [[gamma function]] is a function that satisfies the functional equation <math>f (x + 1) = x f (x)</math> and the initial value <math>f (1) = 1.</math> There are many functions that satisfy these conditions, but the gamma function is the unique one that is [[meromorphic function|meromorphic]] in the whole complex plane, and [[logarithmically convex function|logarithmically convex]] for {{mvar|x}} real and positive ([[Bohr–Mollerup theorem]]).