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The following paper brings an application of the Fourier transformation. I would like to evaluate

$$ G_{k}=\oint_{0}^{2\pi}G(\theta)e^{jk\theta}d\theta, $$

for $k=1,2,...,6$, where

$$ G(\theta)=\frac{1}{F(z)},\qquad z=e^{j\theta}. $$

Suppose that $z\equiv y$, $x=Ry+h$, $h$ is the center of interval, $R$ its size and

$$ F(z)\equiv F(y)=f(Ry+h), $$

where $f(x)=x\tan x-1$, $R=4$, $h=4$, $\theta$ is discretized into 256 equidistant points. Unfortunately, the following Matlab code:

R = 4
h = 4
tmin = 0.000;
tmax = 2 * pi - 0.001;
sampling_rate = 1/256; 
t = (tmin:sampling_rate * (tmax - tmin):tmax)';  
y = R * exp(i * t) + h ;
F = y .* tan(y) - 1;
G = 1.0 ./ F;
Gk = fft(G, 6);

brings different numerical values of $G_{k}$:

$G_{1}=-0.1076-0.1699j$ , $G_{2}=0.0557+0.0357j$, $G_{3}=0.0184+0.0330j$, $G_{4}=-0.0001+0.0327j$, $G_{5}=-0.0185+0.0330j$, $G_{6}=-0.0562+0.0353j$.

Do I understand the problem correctly? Thanks for your help.

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