Asides

Notatka na marginesie

exp(x) = sin(x)

Some function
f(x) = e^x-\sin(x)
gives us interesting properties:

  • on the left side (x<0) we have f(x) \approx \sin(x) and having infinitely many zeros
  • on the right side (x>0) we have f(x) \approx e^x and growing exponentially

However, there is POI about zeros of this function:

exsinx




Since \sin(x) \approx x when x is small, we have approximation of first root as an solution of e^x = - \pi - x
x_1 \approx - \pi - e^{-\pi - e^{-\pi - e^{-\pi - \ldots}}} = -W(e^{-\pi})-\pi \approx -3.18305
where W(x) is Lambert-W function.
The same way we have approximation of second root as an solution of e^x = 2\pi + x x_2 \approx - 2\pi + e^{-2\pi + e^{-2\pi + e^{-2\pi + \ldots}}} = -W(e^{-2\pi})-2\pi \approx -6.28131
We know that e^x \rightarrow 0 as x \rightarrow - \infty there is x_n \rightarrow -n\pi, although we don’t know the exact solution yet..