Symmetry in motion in euler's formula and its breakdown in hyperbolic function and growth function

This study was designed to investigate the symmetry in motion in Euler's formula and its breakdown in hyperbolic function and growth function. The results obtained were as follows. (1) The symmetry in motion between θ and exp(iθ) was shown when θ was kept in motion. The symmetry breakdown in motion between θ and cosh(θ), between θ and sinh(θ), and between θ and exp(θ) was shown when θ was kept in motion. (2) There were some properties that were common to these three functions. (3) If hyperbolic function related exponential function with Lorentz transformation, then the Bondi-k factor was an exponential function. (4) Mathematical relationships between the Bondi-k factor and the imaginary unit were suggested. (5) Mathematical relationships between Euler's formula and growth function were discussed by relating qualitative differences in complex numbers with quantitative differences in real numbers. It was suggested that the breakdown of symmetry in motion in Euler's formula gave hyperbolic function and growth function, where some properties common to these three functions were also observed.