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Euler's formula expresses the rotating complex exponential $e^{-2\pi i ft}$ as $\cos(2\pi ft) - i\sin(2\pi ft)$. When integrated against a signal $g(t)$, this separates the Fourier transform into two distinct real-valued integrals: the real part corresponds to the correlation with cosine, and the imaginary part corresponds to the correlation with negative sine.

Conditions: Using the convention where negative exponents denote clockwise rotation.; $g(t)$ is a real-valued function.

Euler's formula expresses the rotating complex exponential $e^{-2\pi i ft}$ as $\cos(2\pi ft) - i\sin(2\pi ft)$. When integrated against a signal $g(t)$, this separates the Fourier transform into two distinct real-valued integrals: the real part corresponds to the correlation with cosine, and the imaginary part corresponds to the correlation with negative sine.

Conditions: Using the convention where negative exponents denote clockwise rotation.; $g(t)$ is a real-valued function.