**Contents:**

**See also**: Einstein Summation.

## What is Summation Notation?

In calculus, **summation notation** or sigma (Σ) represents **adding many values together**.

The “a_{i}” in the above sigma notation is saying that you sum all of the values of “a”. In other words, you’re adding up a series of values: a_{1}, a_{2}, a_{3}, …, a_{x}.

*i*is the**index of summation.**It doesn’t have to be “i”: it could be any variable (j, k, x etc.).- a
_{i}is theterm in the sum.*i*th *n*and*1*are the**upper and lower bounds**of summation. I’m using “1” here as an example: the lower bound could be an integer less than or equal to*n*.

The following image shows sigma notation for adding up a series of digits from 1 to 6. The lower bound (1) and upper bound (6) are below and above the sigma, respectively. Basically, you start adding at 1 and stop when you get to 6:

In the following example, “k” is the index of summation because there’s a “k” in the formula.

It’s telling you to start at k = 1 (lower bound) and keep on summing. It might seem that you keep on adding infinitely, but you’ll usually stop when your function either converges (settles on a certain number) or clearly diverges (shows no hope of convergence).

## Using Summation Notation: Calculus Example (Rectangles)

Sigma notation can be used in calculus to evaluate **sums of rectangular areas**. You can think of the bounds of summation here as where your rectangles start, and where they end.

**Example problem:** Evaluate the sum of the rectangular areas in the figure below. Use sigma notation:

Step 1: **Multiply the lengths** of the base **by the height** of each rectangle.

- 1 *
^{1}⁄_{3}=^{1}⁄_{3} - 1 *
^{1}⁄_{4}=^{1}⁄_{4} - 1 *
^{1}⁄_{5}=^{1}⁄_{5}

Step 2: **Add up the numbers **you calculated in Step 1:

^{1}⁄_{3} + ^{1}⁄_{4} + ^{1}⁄_{5} = ^{47}⁄_{60}.

Step 3: **Write the summand** ^{1}⁄_{k} to the right of the sigma. The variables i, j, and k are usually used instead of x:

Step 4: **Write the place** where the summation ends at the top of Σ. This is a right-hand Riemann sum and so the measurement ends at the right of the last rectangle, at x = 5.

Step 5: **Write the place where the summation starts **at the bottom of Σ, after the index of summation (in this case, the index of summation is k).

*That’s it!*

## What is a Sigma Function?

The “sigma function” may refer to:

## As a Sum of Positive Divisors

In number theory, the Sigma Function (denoted σ(n) or Σ(n)) of a positive integer is the sum of the positive divisors of n. For example, the number 3 has two positive divisors (1, 3) with a sum of 1 + 3 = 4. So:

σ(3) = 4.

A few more examples:

- σ(6) = (1 + 2 + 3) = 6,
- σ(12) = (1 + 2 + 3 + 4 + 6) = 16,
- σ(15) = (1 + 3 + 5) = 9.

Interestingly, the sigma function for any prime number is just that number plus one. That’s because primes are only divisible by itself and one.

## The Weierstrass Sigma Function

The **Weierstrass sigma function**, usually denoted σ(z), is used in complex analysis and elliptic function theory (an elliptic function is a doubly periodic function). It was named after German mathematician Karl Weierstrass (October 1815 – 19 February 1897) and appeared in several of his works.

The Weierstrass elliptic functions, which includes sigma, are in a relatively simple form and are also called *p-functions*, with a stylized letter P: ℘.

The Weiestrass Sigma function is defined as:

## Properties of the Weierstrass Sigma Function

- σ(z) is an odd function of z. Odd functions are symmetrical about the origin.
- σ(z) is an entire function.

## References

Caldwell, C. Retrieved November 30, 2019 from: https://primes.utm.edu/glossary/page.php?sort=SigmaFunction

Gaberdiel, J. A Study of Perfect Numbers and Related Topics, With Special Emphasis on the Search for an Odd Perfect Number. Retrieved November 30, 2019 from: https://www.math.arizona.edu/~rta/001/gaberdiel/

Komeda, J. et al. The Sigma Function for Weierstrass Semigroups <3, 7, 8> and <6, 13, 14, 15, 16>. International Journal of Mathematics, Vol. 24, No. 11.

Korn, G. & Korn, T. (2013). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. Courier Corporation.

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