moving trig function from the exponent?

Question

Is there a form of $e^{\sin(x)}$ that does not have trig functions in the exponent? I've seen the "Euler Identity" form and looked into series expansions. (thanks for the answers -- sorry I didn't give background earlier.) I'm curious if there was something computationally faster than sin() then exp().

Answer 1

You could use this $$ e^{\sin(x)} = e^{\frac{1}{2i}(e^{ix} - e^{-ix})} $$ But I doubt that's what you're looking for. A common trick in situations like this is to use the taylor series. So you could try $$ e^{\sin(x)} = e^{x-x^3/6 + x^5/120 - \cdots} $$ or $$ e^{\sin(x)} = 1 + \sin(x) + \sin^2(x)/2 + \sin^{3}(x)/6 + \cdots $$