Gauss summation formula

In summary, the Gauss summation formula, also known as the Gauss's trick, is a mathematical formula that allows for the quick calculation of the sum of a series of numbers in a sequence. It was developed by the famous mathematician Carl Friedrich Gauss. The formula works by breaking down the series of numbers into two smaller sequences, one in ascending order and the other in descending order. By adding these two sequences together, all the intermediate terms cancel out, leaving only the first and last terms. The formula then simply multiplies the sum of the two sequences by half the number of terms in the original series to get the final sum. The Gauss summation formula has many practical applications in various fields such as physics, engineering, and computer science.
  • #1
Ted123
446
0
Why is the Gauss summation formula for complex parameters a,b,c: [tex]\displaystyle _2 F_1 (a,b;c;1) = \frac{\Gamma (c) \Gamma (c-a-b)}{\Gamma (c-a) \Gamma (c-b)}[/tex] only valid for [itex]\text{Re}(c-a-b)>0,\;c\neq 0,-1,-2,-3,...[/itex]?
 
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  • #2
It makes sense that [itex]c\neq 0,-1,-2,-3,...[/itex] so that [itex]\Gamma (c)[/itex] is defined but why does [itex]\text{Re}(c-a-b)[/itex] have to be positive?
 

What is the Gauss summation formula?

The Gauss summation formula, also known as the Gauss's trick, is a mathematical formula that allows for the quick calculation of the sum of a series of numbers in a sequence. It was developed by the famous mathematician Carl Friedrich Gauss.

How does the Gauss summation formula work?

The formula works by breaking down the series of numbers into two smaller sequences, one in ascending order and the other in descending order. By adding these two sequences together, all the intermediate terms cancel out, leaving only the first and last terms. The formula then simply multiplies the sum of the two sequences by half the number of terms in the original series to get the final sum.

What is the significance of the Gauss summation formula?

The Gauss summation formula has many practical applications in various fields such as physics, engineering, and computer science. It can be used to efficiently calculate the area under a curve, the sum of a finite arithmetic or geometric series, and even the value of certain integrals. It is also a fundamental concept in the study of calculus and infinite series.

Can the Gauss summation formula be used for any series of numbers?

The Gauss summation formula can only be used for finite arithmetic and geometric series. It cannot be applied to infinite series or series with non-constant differences between terms. However, there are generalizations of the formula that can be used for other types of series.

Are there any limitations to the Gauss summation formula?

While the Gauss summation formula is a powerful tool for calculating series sums, it does have its limitations. It is not applicable to series with non-constant differences between terms, and it cannot be used to calculate sums of infinite series. Additionally, it may not always be the most efficient method of finding the sum of a series, as certain series may have simpler methods of calculation.

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