Fourier Integral¶

\begin{matrix} \int_{- \infty}^{\infty} | f (x) | d x \leftarrow converge \end{matrix}

converges, then

f (x) = \int_{0}^{\infty} [A (ω) \cos (ω x) + B (ω) \sin (x)] d ω

\begin{array}{r} A (ω) = \frac{1}{π} \int_{- \infty}^{\infty} f (x) \cos (ω x) d x B (ω) = \frac{1}{π} \int_{- \infty}^{\infty} f (x) \sin (ω x) d x \end{array}

f (x) = \int_{0}^{\infty} A (ω) \cos (ω x) d ω

\begin{array}{r} A (ω) = \frac{2}{π} \int_{0}^{\infty} f (x) \cos (ω x) d x \end{array}

\begin{array}{r} f (x) = \int_{0}^{\infty} B (ω) \sin (ω x) d ω \\ B (ω) = \frac{2}{π} \int_{0}^{\infty} f (x) \sin (ω x) d x \end{array}

\begin{array}{r} f (x) = \int_{- \infty}^{\infty} C (ω) e^{i ω x} d ω \\ C (ω) = \frac{1}{2 π} \int_{- \infty}^{\infty} f (x) e^{- i ω x} d x \end{array}