In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series

$\sum _{n=-\infty }^{\infty }a_{n}\cdot z^{n}$

then its multisection is a power series of the form

where $\omega =e^{\frac {2\pi i}{q}}$ is a primitive q-th root of unity. This expression is often called a root of unity filter. This solution was first discovered by Thomas Simpson.^{[1]} This expression is especially useful in that it can convert an infinite sum into a finite sum. It is used, for example, in a key step of a standard proof of Gauss's digamma theorem, which gives a closed-form solution to the digamma function evaluated at rational values p/q.

Examples

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Bisection

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In general, the bisections of a series are the even and odd parts of the series.

Higher order multisections are found by noting that all such series must be real-valued along the real line. By taking the real part and using standard trigonometric identities, the formulas may be written in explicitly real form as

^Simpson, Thomas (1757). "CIII. The invention of a general method for determining the sum of every 2d, 3d, 4th, or 5th, &c. term of a series, taken in order; the sum of the whole series being known". Philosophical Transactions of the Royal Society of London. 51: 757–759. doi:10.1098/rstl.1757.0104.