IOD means integration over differentiation. Separately, the first implies discrete sum of infinitesimals ∑���� while the second implies ratio of infinitesimals ����/����. If the limit of ���� exists then the infinitesimals become derivatives and the notation ���� becomes the exact differential ���� or the partial differential �� and both integration and differentiation are transformed from relative discreteness into absolute continuity. On the other hand, a derivative can only exist if and only if the function involving this derivative is continuous. That is to say the limits from the left and the right is the same. However, for infinitely oscillating differentiability, four nonlinear functions; the real exponential, the real sine, the complex exponential, and the complex sine all satisfy conditions of infinitely oscillating differentiation and integration.
Fortunately, more than two hundred twenty five years ago Leonhard Euler (1707-83) had already established the logical mathematical connection between the complex exponential function and the real sine function in his two famous equations: ������(����)=������(��)+��������(��) and ������(-����)=������(��)-��������(��). For ��=�� then ������(����)= ������(-����)=-1 by simply applying the trigonometric fact that ������(��)= ������(-��). Subsequently, the 1st derivative of ������(����)=�� ������(����), the 2nd derivative is ��² ������(����), the 3rd is �� ³������(����), the 4th is �� ⁴������(����), the 5th is ��⁵ ������(����), the 6th is ��⁶ ������(����), the 7th is ��⁷ ������(����), and the 8th is �� ⁸������(����). These represent infinite differentiability while the powers of the imaginary unity represent infinite number of oscillations between real and imaginary factors: ��, ��²=-1, ��³= -��, ��⁴=1, ��⁵=��, ��⁶= -1, ��⁷= -��, and ��⁸=1. These clearly show that the four components repeating real-imaginary cycle is (��,-1,-��,1). Nevertheless, the real exponential function ������(±��) or the complex exponential function ������(±����) are implemented in both quantum mechanics and quantum field theories as the phase factors representing gauge invariance. However, for a quantum theory of space-time, the real exponential function is ������(±ℋ) and the complex exponential function is ������(±��ℋ), where ℋ are symmetric singular Hadamard matrices.