Difference between revisions of "Discontinuous multiplier"
From Encyclopedia of Mathematics
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A quantity depending on one or more parameters and taking two (or more) values. For example, | 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|Dirichlet discontinuous multiplier]], the Dirac [[Delta-function|delta-function]], etc. | 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|Dirichlet discontinuous multiplier]], the Dirac [[Delta-function|delta-function]], etc. | ||
Latest revision as of 19:35, 5 June 2020
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.
How to Cite This Entry:
Discontinuous multiplier. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discontinuous_multiplier&oldid=46728
Discontinuous multiplier. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Discontinuous_multiplier&oldid=46728
This article was adapted from an original article by K.Yu. Bulota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article