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Difference between revisions of "Discontinuous multiplier"

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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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033010/d0330101.png" /></td> </tr></table>
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$$
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{
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\frac{1}{2 \pi i }
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}
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\int\limits _ {2 + i \infty } ^ { {2 }  - i \infty }
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\frac{y ^ {s + 2k }  ds }{s ( s + 1) \dots ( s + 2k) }
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=
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$$
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$$
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= \
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\left \{
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\begin{array}{ll}
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\frac{( y - 1)  ^ {2k} }{( 2k)! }
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  & \textrm{ if }  y \geq  1, k > 0,  \\
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0  & \textrm{ if }  0 \leq  y < 1. \\
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\end{array}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033010/d0330102.png" /></td> </tr></table>
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\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
This article was adapted from an original article by K.Yu. Bulota (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article