The Weierstrass elliptic functions, which includes sigma, are in a relatively simple form and are also called p-functions, with a stylized letter P: ℘. It was named after German mathematician Karl Weierstrass (October 1815 – 19 February 1897) and appeared in several of his works. 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). That’s because primes are only divisible by itself and one. Interestingly, the sigma function for any prime number is just that number plus one. For example, the number 3 has two positive divisors (1, 3) with a sum of 1 + 3 = 4. Sigma notation is named based on its use of the capital Greek letter sigma: When used in the context of mathematics, the capital sigma indicates that something (usually an expression) is being summed. 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. Instead, a method of denoting series, called sigma notation, can be used to efficiently represent the summation of many terms. The sum of positive divisors of some number n, or.Each term is evaluated, then we sum all the values, beginning with the value when i 1 i 1 and ending with the value when i n i n. where ai a i describes the terms to be added, and the i i is called the index. 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). Typically, sigma notation is presented in the form. This is a right-hand Riemann sum and so the measurement ends at the right of the last rectangle, at x = 5. Step 4: Write the place where the summation ends at the top of Σ. The variables i, j, and k are usually used instead of x: Step 3: Write the summand 1⁄ k to the right of the sigma. Step 2: Add up the numbers you calculated in Step 1: Step 1: Multiply the lengths of the base by the height of each rectangle. You can think of the bounds of summation here as where your rectangles start, and where they end.Įxample problem: Evaluate the sum of the rectangular areas in the figure below. Sigma notation can be used in calculus to evaluate sums of rectangular areas. Using Summation Notation: Calculus Example (Rectangles) 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). The symbol (capital sigma) is often used as shorthand notation to indicate the sum of a number of similar terms. It’s telling you to start at k = 1 (lower bound) and keep on summing. In the following example, “k” is the index of summation because there’s a “k” in the formula. Basically, you start adding at 1 and stop when you get to 6: The lower bound (1) and upper bound (6) are below and above the sigma, respectively. The following image shows sigma notation for adding up a series of digits from 1 to 6. I’m using “1” here as an example: the lower bound could be an integer less than or equal to n.
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