I have a very long function for $\{x,b\}\in\mathbb{R}$ and $n=\{1,2,3,4,5,6,7,8,9,10\}$; here, I have only mentioned a short part of that. The function is a sum of the complex exponentials (a picture is attached below).
I want to simplify such a function in the shortest and simplest form possible; I am somehow sure that there should be a multiplicative factor $i$ in the whole function.
My question
What is the most efficient/optimal way to simplify the given function? Does the code
exp // ExpandAll // ExpToTrig // TrigExpand // TrigFactor // Simplify[#, Assumptions -> x > 0 && x \[Element] Reals && b \[Element] Reals && n \[Element] Integers && 0 < n < 11 ] &do the job for me? Since the function is too long, I am afraid of usingFullSimplifydue to timing. In particular, is the assumption $0 < n < 11$ sufficient (since $n=\{1,2,3,4,5,6,7,8,9,10\}$ only)?
I appreciate any comments.
exp := -462 E^( 9 I b + (2 I n \[Pi])/11 + 20 I x) (10 + E^((2 I n \[Pi])/11) + E^((14 I n \[Pi])/11) + 10 E^((16 I n \[Pi])/11) + 7 E^((18 I n \[Pi])/11)) (-1 + E^( 2 I x)) + 66 E^((2 I n \[Pi])/11 + 9 I (b + 2 x)) (-3 + 5 E^((2 I n \[Pi])/11) + 5 E^((14 I n \[Pi])/11) - 3 E^((16 I n \[Pi])/11) + 9 E^((18 I n \[Pi])/11)) (-1 + E^(6 I x)) + 55 E^(9 I b + (2 I n \[Pi])/11 + 16 I x) (7 - 3 E^((2 I n \[Pi])/11) - 3 E^((14 I n \[Pi])/11) + 7 E^((16 I n \[Pi])/11) + 31 E^((18 I n \[Pi])/11)) (-1 + E^( 10 I x)) - E^(9 I b + (2 I n \[Pi])/11 + 14 I x) (319 - 55 E^((2 I n \[Pi])/11) + 4749 E^((4 I n \[Pi])/11) + 4749 E^((12 I n \[Pi])/11) - 55 E^((14 I n \[Pi])/11) + 319 E^((16 I n \[Pi])/11) + 1903 E^((18 I n \[Pi])/11)) (-1 + E^(14 I x)) + E^(11 I b + (2 I n \[Pi])/11 + 12 I x) (963 + 147 E^((2 I n \[Pi])/11) - 11 E^((4 I n \[Pi])/11) - 11 E^((16 I n \[Pi])/11) + 147 E^((18 I n \[Pi])/11)) (-1 + E^(18 I x)) - E^(9 I b + (2 I n \[Pi])/11 + 10 I x) (37 - E^((2 I n \[Pi])/11) - E^((14 I n \[Pi])/11) + 37 E^((16 I n \[Pi])/11) + 253 E^((18 I n \[Pi])/11)) (-1 + E^( 22 I x)) - 528 E^(12 I b + (13 I n \[Pi])/11 + 19 I x) (5 Cos[(n \[Pi])/11] + Cos[(3 n \[Pi])/11]) (-1 + E^( 4 I x)) - 528 E^(10 I b + I n \[Pi] + 19 I x) (3 Cos[(n \[Pi])/11] + 2 Cos[(3 n \[Pi])/11] + Cos[(5 n \[Pi])/11]) (-1 + E^(16 I x)) - E^(11 I b + (2 I n \[Pi])/11 + 14 I x) (1903 + 2 E^((10 I n \[Pi])/ 11) (4749 Cos[(4 n \[Pi])/11] - 55 Cos[(6 n \[Pi])/11] + 319 Cos[(8 n \[Pi])/11])) (-1 + E^(14 I x)) 
TrigToExp. That is a natural way of writing it, because the Fourier basis is linearly independent. I guess more simplifications are possible in principle, maybe one canFactorsomehow, but it would seem that such further simplifications depend on knowing all terms, not just the selection that you have given. $\endgroup$