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f(x) = sin( x * PI / 5 )

(1/10) * exp( 2 * PI * i * 1/10 ) * (0 - 5i) = (0.293893, -0.404508i)

(1/10) * exp( 2 * PI * i * 9/10 ) * (0 + 5i) = (0.293893, 0.404508i)

$$ f(x) = \sin \left( \tfrac{\pi}{5} x \right) $$

$$ \tfrac{1}{10} e^{i 2 \pi (1/10)} \cdot (0 - 5i) = (0.293893 -0.404508i) $$

$$ \tfrac{1}{10} e^{i 2 \pi (9/10)} \cdot (0 + 5i) = (0.293893 +0.404508i) $$

Great to have these expert answers here! I can contribute the 3D plot which supports my intuition and therefore might also be helpful for others. It shows F(t) and how it travels over the original data points (crosses). In order to achieve this, it makes use of the whole space including the complex plane.

Great to have these expert answers here! I can contribute the 3D plot which supports my intuition and therefore might also be helpful for others. It shows f(t) computed by IDFT and how it travels over the original data points (crosses). In order to achieve this, it makes use of the whole space including the complex plane.

EDIT2

Great to have these expert answers here! I can contribute the 3D plot which supports my intuition and therefore might also be helpful for others. It shows F(t) and how it travels over the original data points (crosses). In order to achieve this, it makes use of the whole space including the complex plane.