TY - JOUR

T1 - Introducing complex numbers into basic growth functions - (V) hypothetic factors influencing increase in θ in exp(θ)

AU - Shimojo, Masataka

AU - Ikeda, Kentaro

AU - Ishiwaka, Reiko

AU - Sato, Hiroyuki

AU - Asano, Yoki

AU - Tobisa, Manabu

AU - Nakano, Yutaka

AU - Ohba, Noriko

AU - Eguchi, Minako

AU - Masuda, Yasuhisa

PY - 2005/2

Y1 - 2005/2

N2 - 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.

AB - 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.

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U2 - 10.5109/4631

DO - 10.5109/4631

M3 - Article

AN - SCOPUS:20144380556

SN - 0023-6152

VL - 50

SP - 135

EP - 139

JO - Journal of the Faculty of Agriculture, Kyushu University

JF - Journal of the Faculty of Agriculture, Kyushu University

IS - 1

ER -