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Say I have a series of $n$ complex numbers of the form $A_k e^{(I \ \theta_k x)} $ where $A_k$ is a real number and so is $\theta_k$ and $k$ runs from $1$ to $n$. $x$ is an algebraic symbol.

Example I have to calculate:-

$ 3e^{5Ix}+10e^{7Ix}+2e^{1Ix}$

Now I need to find the summation of the series. I can do this manually because I know all the phase angles (in terms of x) and the lengths of the complex numbers in space, so I will simply add them up like vectors using the parallelogram law and apply the appropriate formulas to successfully reduce the sum of the series to one number say $R e^{I\ F}$ where R and F will be functions in $x$. How do I calculate this number with Mathematica?

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  • $\begingroup$ Did you try just typing the sum into Mathematica and evaluating it? Is that result inadequate in some way? $\endgroup$
    – David Z
    Commented Feb 3, 2013 at 8:48
  • $\begingroup$ When I Type say : 3 Exp[5 x i] + 10 Exp[7 x i] + 2 Exp[1 x i] , it just gives me back the same thing in standard form. $\endgroup$
    – swanar
    Commented Feb 3, 2013 at 8:54
  • $\begingroup$ OK, but then what result are you looking for? What sort of simplification do you expect Mathematica to make? (Note that this sum can't be written as a single term of the form $Re^{i\theta x}$, if that's what you had in mind.) $\endgroup$
    – David Z
    Commented Feb 3, 2013 at 8:56
  • $\begingroup$ The simplification will be such that finally the answer should be in $R e^{i F}$ form. R and F will be functions in x. Thankyou for correcting I have modified the original problem a bit. $\endgroup$
    – swanar
    Commented Feb 3, 2013 at 9:00
  • $\begingroup$ Do you really use lowercase 'i'? That should be I or esc i esc. $\endgroup$ Commented Feb 3, 2013 at 9:04

2 Answers 2

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number = 3 E^(5 I x) + 10 E^(7 I x) + 2 E^(1 I x);
abs = ComplexExpand[Abs[number], TargetFunctions -> {Re, Im}];
arg = ComplexExpand[Arg[number], TargetFunctions -> {Re, Im}];
abs E^(I arg)

it gives : $e^{i \text{ArcTan}[2 \text{Cos}[x]+3 \text{Cos}[5 x]+10 \text{Cos}[7 x],2 \text{Sin}[x]+3 \text{Sin}[5 x]+10 \text{Sin}[7 x]]} \surd \left((2 \text{Cos}[x]+3 \text{Cos}[5 x]+10 \text{Cos}[7 x])^2+(2 \text{Sin}[x]+3 \text{Sin}[5 x]+10 \text{Sin}[7 x])^2\right)$

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You could do :

points = {3 Exp[5 x I], 10 Exp[7 x I], 2 Exp[1 x I]};
eqn = ComplexExpand[Total[points] - r Exp[I f], TargetFunctions -> {Re, Im}];
cond = r >= 0;

sol = Reduce[Thread[And[Thread[Simplify[{Re[eqn], Im[eqn]}, 
        Assumptions -> {x \[Element] Reals, r \[Element] Reals, f \[Element] Reals}] == 0], cond]], 
       {r, f}, Reals];

(sol /.  {C[1] -> 0, x -> Pi/4})
(* r == Sqrt[101] && 
   f == -2 ArcTan[Sqrt[(3/Sqrt[2] - 6 Sqrt[2] + Sqrt[101])/(-(3/Sqrt[2]) + 6 Sqrt[2] + 
        Sqrt[101])]] *)
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