Given the function $$f(x)= e^{\sin{x}}$$ I have to write it without using the exponential or sine function.
I came to this point $$f(x) = \sum_{k=0}^{\infty} \frac{\sin^k{x}}{k!}$$
How can I get rid of that $\sin^k(x)$?
Thanks in advance!
Asked
Active
Viewed 104 times
3
-
put the taylor series for sin(x) – happymath Mar 09 '14 at 08:38
-
@user129901 that will get very messy though. – Guy Mar 09 '14 at 08:45
-
@Sabyasachi or else first do taylor expansion of sin(x) and then use taylor series of $e^x$ this looks like a nice product – happymath Mar 09 '14 at 08:47
-
@user129901 isn't it the same? What would be the difference between two methods? – Simon Mar 09 '14 at 08:50
-
@Simon both will give same series but taking an infinite series to some power is definitely less preferable than easily writing out the product for example $e ^{x-x^2}$ is $(1+x+x^2/2!..)$$(1-x^2+x^4/2!..)$ – happymath Mar 09 '14 at 08:53
-
@Simon the final answer would be the same if you do it right, but using the Taylor series for $e^x$ first, will lead to a messy calculation. – Guy Mar 09 '14 at 08:53
-
See the formula in my answer. – Mhenni Benghorbal Mar 09 '14 at 08:57