Discontinuous multiplier

A quantity depending on one or more parameters and taking two (or more) values. For example,

$$ { \frac{1}{2 \pi i } } \int\limits _ {2 + i \infty } ^ { {2 } - i \infty } \frac{y ^ {s + 2k } ds }{s ( s + 1) \dots ( s + 2k) } = $$

$$ = \ \left \{ \begin{array}{ll} \frac{( y - 1) ^ {2k} }{( 2k)! } & \textrm{ if } y \geq 1, k > 0, \\ 0 & \textrm{ if } 0 \leq y < 1. \\ \end{array} \right .$$

Discontinuous multipliers are applied to make a formal extension of the domain of summation or integration, or to reduce a given expression to another to which given formulas or transformations can be applied. Other examples are the Dirichlet discontinuous multiplier, the Dirac delta-function, etc.