Difference between revisions of "Cyclic sum"
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==Rigorous definition== | ==Rigorous definition== | ||
− | Consider a function <math>f(a_1,a_2,a_3,\ldots a_n</math>. The cyclic sum <math>\ | + | Consider a function <math>f(a_1,a_2,a_3,\ldots a_n)</math>. The cyclic sum <math>\sum_{cyc} f(a_1,a_2,a_3,\ldots a_n)</math> is equal to |
<cmath>f(a_1,a_2,a_3,\ldots a_n)+f(a_2,a_3,a_4,\ldots a_n,a_1)+f(a_3,a_4,\ldots a_n,a_1,a_2)\ldots+f(a_n,a_1,a_2,\ldots a_{n-1})</cmath> | <cmath>f(a_1,a_2,a_3,\ldots a_n)+f(a_2,a_3,a_4,\ldots a_n,a_1)+f(a_3,a_4,\ldots a_n,a_1,a_2)\ldots+f(a_n,a_1,a_2,\ldots a_{n-1})</cmath> | ||
− | Note that not all permutations of the | + | Note that not all permutations of the variables are used; they are just cycled through. |
==Notation== | ==Notation== | ||
− | + | A cyclic sum is often specified by having the variables to cycle through underneath the sigma, as follows: <math>\sum_{a,b,c}\frac{ab}{cd}</math>. Note that a cyclic sum need not cycle through all of the variables. | |
+ | |||
+ | A cyclic sum is also sometimes specified by <math>\sum_{cyc}</math>. This notation implies that all variables are cycled through. | ||
==See also== | ==See also== | ||
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*[[Symmetric sum]] | *[[Symmetric sum]] | ||
+ | [[Category:Algebra]] | ||
[[Category:Definition]] | [[Category:Definition]] |
Revision as of 13:20, 14 July 2021
A cyclic sum is a summation that cycles through all the values of a function and takes their sum, so to speak.
Rigorous definition
Consider a function . The cyclic sum is equal to
Note that not all permutations of the variables are used; they are just cycled through.
Notation
A cyclic sum is often specified by having the variables to cycle through underneath the sigma, as follows: . Note that a cyclic sum need not cycle through all of the variables.
A cyclic sum is also sometimes specified by . This notation implies that all variables are cycled through.