In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.

Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem.

Suppose that ${\displaystyle A(x)}$ and ${\displaystyle B(x)}$ are two complex-valued functions on ${\displaystyle \mathbb {R} }$ of period ${\displaystyle 2\pi }$ that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series

$${\displaystyle A(x)=\sum _{n=-\infty }^{\infty }a_{n}e^{inx}}$$ and
$${\displaystyle B(x)=\sum _{n=-\infty }^{\infty }b_{n}e^{inx}}$$

respectively. Then

${\displaystyle \sum _{n=-\infty }^{\infty }a_{n}{\overline {b_{n}}}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }A(x){\overline {B(x)}}\,\mathrm {d} x,}$   ()

where ${\displaystyle i}$ is the imaginary unit and horizontal bars indicate complex conjugation.

Substituting ${\displaystyle A(x)}$ and ${\displaystyle {\overline {B(x)}}}$ in the integral: