I was reading some stuff earlier today, and I wasn't sure how they changed the exponentials to trigs in this expression:
$$C_1x^{-1/4}\exp\left(\frac{2}{3}ix^{3/2}\right)+C_2x^{-1/4}\exp\left(-\frac{2}{3}ix^{3/2}\right)$$
$=$
$$A_1x^{-1/4}\sin\left(\frac{2}{3}x^{3/2}\right)+A_2x^{-1/4}\cos\left(\frac{2}{3}x^{3/2}\right)$$
Does it have something to do with euler's formula?
And can I change $$C_1x^{-1/2}\exp\left(\frac{i}{2x^2}\right)+C_2x^{-1/2}\exp\left(-\frac{i}{2x^2}\right)$$ to $$A_1x^{-1/2}\sin\left(\frac{1}{2x^2}\right)+A_2x^{-1/2}\cos\left(\frac{1}{2x^2}\right)?$$