Editing Arithmetic–geometric mean (section)

==Properties==
Both the geometric mean and arithmetic mean of two positive numbers {{mvar|x}} and {{mvar|y}} are between the two numbers. (They are ''strictly'' between when {{math|''x'' ≠ ''y''}}.) The geometric mean of two positive numbers is [[Inequality of arithmetic and geometric means|never greater than the arithmetic mean]].<ref>{{cite book |last=Bullen |first=P. S. |contribution=The Arithmetic, Geometric and Harmonic Means |date=2003 |url=http://link.springer.com/10.1007/978-94-017-0399-4_2 |title=Handbook of Means and Their Inequalities |pages=60–174 |access-date=2023-12-11 |place=Dordrecht |publisher=Springer Netherlands |language=en |doi=10.1007/978-94-017-0399-4_2 |isbn=978-90-481-6383-0}}</ref> So the geometric means are an increasing sequence {{math|''g''{{sub|0}} ≤ ''g''{{sub|1}} ≤ ''g''{{sub|2}} ≤ ...}}; the arithmetic means are a decreasing sequence {{math|''a''{{sub|0}} ≥ ''a''{{sub|1}} ≥ ''a''{{sub|2}} ≥ ...}}; and {{math|''g<sub>n</sub>'' ≤ ''M''(''x'', ''y'') ≤ ''a<sub>n</sub>''}} for any {{mvar|n}}. These are strict inequalities if {{math|''x'' ≠ ''y''}}.

{{math|''M''(''x'', ''y'')}} is thus a number between {{math|''x''}} and {{math|''y''}}; it is also between the geometric and arithmetic mean of {{math|''x''}} and {{math|''y''}}.

If {{math|''r'' ≥ 0}} then {{math|''M''(''rx'', ''ry'') {{=}} ''r M''(''x'', ''y'')}}.

There is an integral-form expression for {{math|''M''(''x'', ''y'')}}:<ref>{{dlmf|first1=B. C.|last1=Carson|id=19.8.i|title=Elliptic Integrals|mode=cs1}}</ref><math display=block>\begin{align}
 M(x,y) &= \frac{\pi}{2} \left( \int_0^\frac{\pi}{2}\frac{d\theta}{\sqrt{x^2\cos^2\theta+y^2\sin^2\theta}} \right)^{-1}\\
&=\pi\left(\int_0^\infty \frac{dt}{\sqrt{t(t+x^2)(t+y^2)}}\right)^{-1}\\
        &= \frac{\pi}{4} \cdot \frac{x + y}{K\left( \frac{x - y}{x + y} \right)}
\end{align}</math>where {{math|''K''(''k'')}} is the [[elliptic integral|complete elliptic integral of the first kind]]:<math display="block">K(k) = \int_0^\frac{\pi}{2}\frac{d\theta}{\sqrt{1 - k^2\sin^2\theta}} </math>Since the arithmetic–geometric process converges so quickly, it provides an efficient way to compute elliptic integrals, which are used, for example, in [[elliptic filter]] design.<ref name="Dimopoulos2011">{{cite book |author-first=Hercules G. |author-last=Dimopoulos |title=Analog Electronic Filters: Theory, Design and Synthesis |url=https://books.google.com/books?id=6W1eX4QwtyYC&pg=PA147 |year=2011 |publisher=Springer |isbn=978-94-007-2189-0 |pages=147–155 }}</ref>


The arithmetic–geometric mean is connected to the [[Theta function|Jacobi theta function]] <math>\theta_3</math> by<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} pages 35, 40</ref><math display="block">M(1,x)=\theta_3^{-2}\left(\exp \left(-\pi \frac{M(1,x)}{M\left(1,\sqrt{1-x^2}\right)}\right)\right)=\left(\sum_{n\in\mathbb{Z}}\exp \left(-n^2 \pi \frac{M(1,x)}{M\left(1,\sqrt{1-x^2}\right)}\right)\right)^{-2},</math>which upon setting <math>x=1/\sqrt{2}</math> gives<math display="block">M(1,1/\sqrt{2})=\left(\sum_{n\in\mathbb{Z}}e^{-n^2\pi}\right)^{-2}.</math>