Arithmetic–geometric mean

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Arithmetic–geometric mean

The arithmetic–geometric mean (AGM) is a concept in mathematics that represents the average of two numbers, calculated through an iterative process that combines both the arithmetic mean and the geometric mean. This mean is particularly useful in various fields such as number theory, complex analysis, and elliptic integrals.

Definition

The arithmetic–geometric mean of two non-negative numbers a and b is defined through the following iterative process:

1. Set a₀ = a and b₀ = b. 2. Compute the arithmetic mean: aₙ₊₁ = (aₙ + bₙ) / 2. 3. Compute the geometric mean: bₙ₊₁ = √(aₙ * bₙ). 4. Repeat steps 2 and 3 until aₙ and bₙ converge to a common value.

The common value to which the sequences {aₙ} and {bₙ} converge is the arithmetic–geometric mean of a and b, denoted as AGM(a, b).

Properties

• The AGM is symmetric, meaning AGM(a, b) = AGM(b, a).
• It is always between the arithmetic mean and the geometric mean of the two numbers.
• The AGM can be used to compute elliptic integrals and has applications in computational mathematics.

Applications

The arithmetic–geometric mean has several important applications:

• In the computation of elliptic integrals, where it provides an efficient method for numerical evaluation.
• In number theory, particularly in the context of modular functions and theta functions.
• In complex analysis, where it is used to study the properties of elliptic functions.

Related Concepts

• Arithmetic mean
• Geometric mean
• Elliptic integral
• Number theory
• Complex analysis

See Also

• Mean
• Harmonic mean
• Quadratic mean
• Elliptic function