Introducing complex numbers into basic growth functions - (V) hypothetic factors influencing increase in θ in exp(θ)

Masataka Shimojo, Kentaro Ikeda, Reiko Ishiwaka, Hiroyuki Sato, Yoki Asano, Manabu Tobisa, Yutaka Nakano, Noriko Ohba, Minako Eguchi, Yasuhisa Masuda

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3 Citations (Scopus)


The present trial was designed to investigate hypothetic factors influencing the increase in θ after the appearance of exp(θ) from exp(iθ). Hypothetic factors used in this trial were related to properties of indefinite integral and Euler's formula. The results obtained were as follows. It was shown hypothetically that '(-1)+1' appeared from '0' through indefinite integral of f(θ) on condition that f(θ)=0, where there was not an increase in θ. The hypothetic breakdown of multiplication form connecting eight complex numbers to construct '-1' left 2exp(iθ), and one of the two exp(iθ) was changed into exp(θ) by the product of 'iθ 'and '-i' (clockwise π/2 rotation of variable). The periodic increase in θ, the different breakdown of multiplication form and changing one of the complex numbers into real number gave [-exp(i(θ + 2 π))-exp((θ + 2 π)]. Consequently it was shown that exp(iθ) was offset by -exp(i(θ + 2 π)), due to the periodic property. However, both exp(θ) and -exp(θ + 2 π) were left because they were not periodic, where there was an increase in θ if the minus sign was disregarded.

Original languageEnglish
Pages (from-to)135-139
Number of pages5
JournalJournal of the Faculty of Agriculture, Kyushu University
Issue number1
Publication statusPublished - Feb 1 2005


All Science Journal Classification (ASJC) codes

  • Biotechnology
  • Agronomy and Crop Science

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