Remarks About the Square Equation. Functional Square Root of a Logarithm

• Autorzy: Arkadiusz Maciuk , Antoni Smoluk
• Języki publikacji: EN

Abstrakty

EN

The study shows that the functional equation f (f (x)) = ln(1 + x) has a unique result in a semigroup of power series with the intercept equal to 0 and the function composition as an operation. This function is continuous, as per the work of Paulsen [2016]. This solution introduces into statistics the law of the one-and-a-half logarithm. Sometimes the natural growth processes do not yield to the law of the logarithm, and a double logarithm weakens the growth too much. The law of the one-and-a-half logarithm proposed in this study might be the solution.(original abstract)

Słowa kluczowe

EN

Financial mathematics; Log-linear models; Functional equation

PL

Matematyka finansowa; Modele logarytmiczno-liniowe; Równanie funkcyjne

• Czasopismo: Mathematical Economics
• Rocznik: 2016
• Numer: nr 12 (19)
• Strony: 39--52

Twórcy

autor

Arkadiusz Maciuk

• Wrocław University of Economics, Poland

autor

Antoni Smoluk

• Wrocław University of Economics, Poland

Bibliografia

• Bödewadt U.T. (1944). Reeller Funktionen zur Iteration. Math. Z. 49, pp. 497-516.
• Iga K. Continuous half-iterates of functions. Manuscript. URLs or http://math.stan-ford.edu/~iga http://math.pepperdine.edu/kiga/Papers/halfiter.ps.
• Kneser H. (1950). Reelle analytische Lösungen der Gleichung 𝜓𝜓􀵫𝜓𝜓(𝑥𝑥)􀵯=𝑒𝑒𝑥𝑥 und verwand-ter Funktionalgleichungen. Journal für die reine und angewandte Mathematik 187, pp. 56-67.
• KuczmaM. (1969). Fractional iteration of differentiable functions. Annales Polonici Mathe-matici. Vol. 2. No. 22, pp. 217-227.
• Paulsen W. (2016). Finding the natural solution to 𝑓𝑓(𝑓𝑓(𝑥𝑥))=𝑒𝑒𝑥𝑥. The Korean Journal of Mathematics 24 (1), pp. 81-106.
• Zhang W. (1997). PM functions. Their characteristic intervals and iterative roots. Annales Polonici Mathematici. Vol. 2. No. 65, pp. 119-128.
• Zubrzycki S. (1962). Wzór rekurencyjny dla ilości partycji ograniczonych. Applicationes Mathematica 6 (2), pp. 231-234.

Identyfikatory

DOI

10.15611/me.2016.12.04