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Where in the definition of the complex exponential function is π used? IIRC its: the exponential function is defined to be its own derivative (note 1), i is the
by moring 3y ago
Where in the definition of the complex exponential function is π used? IIRC its: the exponential function is defined to be its own derivative (note 1), i is the square root of -1, and exp(ix) is observed to have a period of 2π. There isn't any arbitrary choices in there that could be said to be defined in such a way that π results.
note 1: exp(x) can alternatively be defined by the exponential series, but that series does contain arbitrary numbers that could be said to be selected in such a way that π results.
- nlitened 3y agoAs I understand, "complex exponential" function f(x) = e^ix must satisfy only two equalities: f(0) = 1 f'(x) = i f(x) So any function that satisfies these equalities can work as a "complex exponential" function which we denote as e^ix. So we can define a function with period of 1, and use it everywhere — then "2π" vanishes from most equations, and the complex math still works and all equalities hold.*
- messe 3y agoThe only function that satisfies those two equalities is e^ix which has period 2π.
- nlitened 3y agoHey, you're actually right! My bad
- jojobas 3y agoConsider simultaneous functional equations: f(x) = dg(x)/dx g(x) = df(x)/dx Its only linearly independent solutions are sin(x+c) and cos(x+c) with, x being in radians, periodic by 2pi.