# Simplifying a sum of complex exponential terms

I have this function for $$n=\{1,2,3,4,5,6,7,8,9,10\}$$. Is there any hope to significantly simplify this function (make it shortened) for general $$n$$? I use Simplify and FullSimplify, but it does not help much.

f[n_]:=11 + 2 E^((2 I n \[Pi])/11) + 5 E^((4 I n \[Pi])/11) + 8 E^((6 I n \[Pi])/11) + 14 E^((8 I n \[Pi])/11) + 20 E^((10 I n \[Pi])/11) + 30 E^((12 I n \[Pi])/11) + 22 E^((18 I n \[Pi])/11) + 17 E^((20 I n \[Pi])/11) +  6 E^((24 I n \[Pi])/11) - 2 E^((26 I n \[Pi])/11) -
8 E^((28 I n \[Pi])/11) - 20 E^((30 I n \[Pi])/11) -
31 E^((32 I n \[Pi])/11) - 60 E^((34 I n \[Pi])/11) -
31 E^((40 I n \[Pi])/11) - 20 E^((42 I n \[Pi])/11) -
2 E^((46 I n \[Pi])/11) + 6 E^((48 I n \[Pi])/11) +
10 E^((50 I n \[Pi])/11) + 17 E^((52 I n \[Pi])/11) +
22 E^((54 I n \[Pi])/11) + 30 E^((56 I n \[Pi])/11) +
20 E^((62 I n \[Pi])/11) + 14 E^((64 I n \[Pi])/11) +
5 E^((68 I n \[Pi])/11) + 2 E^((70 I n \[Pi])/11) + E^(( 72 I n \[Pi])/11)



$$f(n)=2 e^{\frac{2 i \pi n}{11}}+5 e^{\frac{4 i \pi n}{11}}+8 e^{\frac{6 i \pi n}{11}}+14 e^{\frac{8 i \pi n}{11}}+20 e^{\frac{10 i \pi n}{11}}+30 e^{\frac{12 i \pi n}{11}}+22 e^{\frac{18 i \pi n}{11}}+17 e^{\frac{20 i \pi n}{11}}+6 e^{\frac{24 i \pi n}{11}}-2 e^{\frac{26 i \pi n}{11}}-8 e^{\frac{28 i \pi n}{11}}-20 e^{\frac{30 i \pi n}{11}}-31 e^{\frac{32 i \pi n}{11}}-60 e^{\frac{34 i \pi n}{11}}-31 e^{\frac{40 i \pi n}{11}}-20 e^{\frac{42 i \pi n}{11}}-2 e^{\frac{46 i \pi n}{11}}+6 e^{\frac{48 i \pi n}{11}}+10 e^{\frac{50 i \pi n}{11}}+17 e^{\frac{52 i \pi n}{11}}+22 e^{\frac{54 i \pi n}{11}}+30 e^{\frac{56 i \pi n}{11}}+20 e^{\frac{62 i \pi n}{11}}+14 e^{\frac{64 i \pi n}{11}}+5 e^{\frac{68 i \pi n}{11}}+2 e^{\frac{70 i \pi n}{11}}+e^{\frac{72 i \pi n}{11}}+11$$

• Is a general n an integer? Or by general do you mean real or even complex?
– Hugh
Commented Nov 7, 2022 at 16:19
• @Hugh $n$ is a positive integer taking the values $1$ to $10$ as I have written in the question. Commented Nov 7, 2022 at 16:21
• it will simplify $e^{\frac{k n }{m} i \pi}$ when $\frac{k n }{m}$ comes out to have special values. These are multiple of integers or multiple half integers. I do not see how it will simplify any more for general cases. Commented Nov 7, 2022 at 16:35
• A more specific title could be helpful to future users searching for a similar question. Perhaps: " Simplifying a sum of complex exponential terms". Maybe more specific titles also lead to better duplicate suggestions by stack exchange when writing the question. Commented Nov 7, 2022 at 17:52
• @userrandrand Thanks; I leave this to you to change the title. Commented Nov 7, 2022 at 21:38

(expression /. {n -> Log[x]/(2*Pi*I/11)}) /.
{x^m_ -> x^Mod[m,11]} //Factor //InputForm
(* 11*(1 + x)*(1 + x^2 + x^4 + x^9) *)

• @charmin Which answer is simpler and are you likely to come up with it yourself and use in the future? Commented Nov 8, 2022 at 1:33
• @charmin Don't sweat it. Your question. Your choice. It's all good. Commented Nov 9, 2022 at 0:08
expression =
11 + 2 E^((2 I n π)/11) + 5 E^((4 I n π)/11) +
8 E^((6 I n π)/11) + 14 E^((8 I n π)/11) +
20 E^((10 I n π)/11) + 30 E^((12 I n π)/11) +
22 E^((18 I n π)/11) + 17 E^((20 I n π)/11) +
6 E^((24 I n π)/11) - 2 E^((26 I n π)/11) -
8 E^((28 I n π)/11) - 20 E^((30 I n π)/11) -
31 E^((32 I n π)/11) - 60 E^((34 I n π)/11) -
31 E^((40 I n π)/11) - 20 E^((42 I n π)/11) -
2 E^((46 I n π)/11) + 6 E^((48 I n π)/11) +
10 E^((50 I n π)/11) + 17 E^((52 I n π)/11) +
22 E^((54 I n π)/11) + 30 E^((56 I n π)/11) +
20 E^((62 I n π)/11) + 14 E^((64 I n π)/11) +
5 E^((68 I n π)/11) + 2 E^((70 I n π)/11) +
E^((72 I n π)/11)


Notice that the argument of every exponential is imaginary which can be checked with:

And @@ (Re[#/n] == 0 &) /@ Cases[expression, Exp[s_] -> s, All]


(* True *)

It is then sufficient to consider the imaginary argument of each exponential modulo $$2\pi$$ which, for integer n, allows the simplification:

Note: below …=\[Ellipsis]

simplified…expression=expression /. Exp[n*v_] :> Exp[n*Mod[v, 2*Pi]]
// FullSimplify


(* 11 (1 + E^((2 I n π)/11)) (1 + E^((4 I n π)/11) + E^(( 8 I n π)/11) + E^((18 I n π)/11)) *)

In LaTex:

$$11 \left(1+e^{\frac{2 i \pi n}{11}}\right) \left(e^{\frac{4 i \pi n}{11}}+e^{\frac{8 i \pi n}{11}}+e^{\frac{18 i \pi n}{11}}+1\right)$$

Verification:

 And @@ Table[
simplified…expression == expression, {n, 0, 10}]


(* True *)