try to integrate Well-known functions in high-school mathematics sin x, cos x ex, log x We try to integrate these functions into a whole with single differential equation: d f (z) f(z), f (0) 1 dz ,where f is defined as complex analytic function f:C->C
Solve! Brief proof df dz log f z C f C is integral constant, then f (z) Aez In remembering f(0)=1, we obtain A=1. In short, f (z) ez Q.E.D.//
Euler identity P utting , we find a kind of compatibility between exponent function and trig function as follows: , where the last equality comes from Euler identity:
Interpretation of differentiation of trig functions Try
to differentiate by variable y,
Comparing the real & imaginary part, we obtain In considering these equations, we only used the differential equation f’=f. i.e. We may interpret that the differentiation of trig functions comes from “imaginary part” of the differentiation of exponent function.