Introducing complex numbers into basic growth functions - (VIII) a series of hypotheses for appearance of exp(θ) from complex representation of '0=1+(-1)' under the concept of symmetry -

Exponential functions with base e are often used for the analysis and estimation of some aspects in ruminant agriculture. The present study was designed to investigate a series of hypotheses influencing the appearance of exp(θ) from '0' using the concept of symmetry. The results obtained were as follows. There was a hypothetic appearance of '1+(-1)' from '0' if there were concepts of function, indefinite integral and symmetry. Then, '1+(-1)' was described using the product of complex numbers, in which this equality held for even when the variable took any value. Therefore, the variable θ in exp(±iθ) was replaced with ∓iθ. This changed exp(iθ) and exp(-iθ) into exp(±θ) and exp(∓θ), respectively, a transformation of variables from imaginary numbers into real numbers where the equality with '1+(-1)' was conserved in the new equation. The hypothetic breakdown of product from in the new equation left exp(±θ), for example. The property that the new equation held for all θ s might give a hypothetic case where θ took values in ascending order, namely an increase in θ. If this hypothetic property was inherited to exp(±θ) that was left, then the growth phenomenon described using the definite integral of exp(θ) might be allowed to occur with an increase in θ, but this speculation will require further investigation.