- Notation
- History
- Solutions
- Examples
- See also
- References
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function {{math| g}} is a function {{math|f}} satisfying {{math|1=f(f(x)) = g(x)}} for all {{math|x}}.
Notation
Notations expressing that {{math|f}} is a functional square root of {{math|g}} are {{math|1=f = g[1/2]}} and {{math|1=f = g1/2}}.
History
- The functional square root of the exponential function (now known as a half-exponential function) was studied by Hellmuth Kneser in 1950.[1]
- The solutions of {{math|1=f(f(x)) = x}} over
(the involutions of the real numbers) were first studied by Charles Babbage in 1815, and this equation is called Babbage's functional equation.[2] A particular solution is {{math|1=f(x) = (b − x)/(1 + cx)}} for {{math|bc ≠ −1}}. Babbage noted that for any given solution {{math|f}}, its functional conjugate {{math|Ψ−1 ○ f ○ Ψ}} by an arbitrary invertible function {{math|Ψ}} is also a solution. In other words, the group of all invertible functions on the real line acts on the subgroup consisting of solutions to Babbage's functional equation by conjugation.
Solutions
A systematic procedure to produce arbitrary functional {{mvar|n}}-roots (including, beyond {{math|1=n = 1/2}}, continuous, negative, and infinitesimal {{mvar|n}}) relies on the solutions of Schröder's equation.[3][4][5]
Examples
- {{math|1=f(x) = 2x2}} is a functional square root of {{math|1=g(x) = 8x4}}.
- A functional square root of the {{mvar|n}}th Chebyshev polynomial, {{math|1=g(x) = Tn(x)}}, is {{math|1=f(x) = cos({{radical|n}} arccos(x))}}, which in general is not a polynomial.
- {{math|1=f(x) = x/({{radical|2}} + x(1 − {{radical|2}}))}} is a functional square root of {{math|1=g(x) = x/(2 − x)}}.
{{math|sin[2](x) {{=}} sin(sin(x))}} [red curve]
{{math|sin[1](x) {{=}} sin(x) {{=}} rin(rin(x))}} [blue curve]
{{math|sin[½](x) {{=}} rin(x) {{=}} qin(qin(x))}} [orange curve]
{{math|sin[¼](x) {{=}} qin(x)}} [black curve above the orange curve]
{{math|sin[–1](x) {{=}} arcsin(x)}} [dashed curve]
(Cf. the general pedagogy web-site.[6] For the notation, see .)
See also
{{Col-begin}}{{Col-1-of-2}}- Iterated function
- Function composition
- Abel equation
- Schröder's equation
- Flow (mathematics)
- Superfunction
- Fractional calculus
- Half-exponential function
References
1. ^{{cite journal|author=Kneser, H. |authorlink=Hellmuth Kneser|title=Reelle analytische Lösungen der Gleichung φ(φ(x)) = ex und verwandter Funktionalgleichungen|journal=Journal für die reine und angewandte Mathematik|url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002175851|volume=187|year=1950|pages=56–67}}
2. ^Jeremy Gray and Karen Parshall (2007) Episodes in the History of Modern Algebra (1800–1950), American Mathematical Society, {{ISBN|978-0-8218-4343-7}}
3. ^{{cite journal |author=Schröder, E. |authorlink=Ernst Schröder |year=1870 |title=Ueber iterirte Functionen|journal=Mathematische Annalen |volume=3 |issue= 2|pages=296–322 | doi=10.1007/BF01443992 | id= |url= |accessdate= |quote= }}
4. ^{{cite journal |author=Szekeres, G.|authorlink=George Szekeres| year=1958|title=Regular iteration of real and complex functions |journal=Acta Mathematica |volume=100|issue=3–4 |pages=361–376 |doi= 10.1007/BF02559539 }}
5. ^{{cite journal |author= Curtright, T.|authorlink= Thomas Curtright| year= 2011|author2=Zachos, C. |authorlink2=Cosmas Zachos|author3=Jin, X. |title=Approximate solutions of functional equations |journal= Journal of Physics A |volume= 44|issue= 40 |pages= 405205|doi=10.1088/1751-8113/44/40/405205|arxiv=1105.3664|bibcode=2011JPhA...44N5205C}}
6. ^Curtright, T.L. Evolution surfaces and Schröder functional methods.
{{DEFAULTSORT:Functional Square Root}}{{mathanalysis-stub}}2. ^Jeremy Gray and Karen Parshall (2007) Episodes in the History of Modern Algebra (1800–1950), American Mathematical Society, {{ISBN|978-0-8218-4343-7}}
3. ^{{cite journal |author=Schröder, E. |authorlink=Ernst Schröder |year=1870 |title=Ueber iterirte Functionen|journal=Mathematische Annalen |volume=3 |issue= 2|pages=296–322 | doi=10.1007/BF01443992 | id= |url= |accessdate= |quote= }}
4. ^{{cite journal |author=Szekeres, G.|authorlink=George Szekeres| year=1958|title=Regular iteration of real and complex functions |journal=Acta Mathematica |volume=100|issue=3–4 |pages=361–376 |doi= 10.1007/BF02559539 }}
5. ^{{cite journal |author= Curtright, T.|authorlink= Thomas Curtright| year= 2011|author2=Zachos, C. |authorlink2=Cosmas Zachos|author3=Jin, X. |title=Approximate solutions of functional equations |journal= Journal of Physics A |volume= 44|issue= 40 |pages= 405205|doi=10.1088/1751-8113/44/40/405205|arxiv=1105.3664|bibcode=2011JPhA...44N5205C}}
6. ^Curtright, T.L. Evolution surfaces and Schröder functional methods.
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