complex arithmetic-geometric mean

It is also possible to define the arithmetic-geometric meanDlmfDlmfMathworldPlanetmath for complex numbersMathworldPlanetmathPlanetmath. To do this, we first must make the geometric meanMathworldPlanetmath unambiguous by choosing a branch of the square rootMathworldPlanetmath. We may do this as follows: Let a and b br two non-zero complex numbers such that asb for any real number s<0. Then we will say that c is the geometric mean of a and b if c2=ab and c is a convex combination of a and b (i.e. c=sa+tb for positive real numbers s and t).

Geometrically, this may be understood as follows: The condition asb means that the angle between 0a and 0b differs from π. The square root of ab will lie on a line bisecting this angle, at a distance |ab| from 0. Our condition states that we should choose c such that 0c bisects the angle smaller than π, as in the figure below:


Analytically, if we pick a polar representation a=|a|eiα, b=|b|eiβ with |α-β|<π, then c=|ab|eiα+β2. Having clarified this preliminary item, we now proceed to the main definition.

As in the real case, we will define sequences of geometric and arithmetic meansMathworldPlanetmath recursively and show that they converge to the same limit. With our convention, these are defined as follows:

g0 =a
a0 =b
gn+1 =angn
an+1 =an+gn2

We shall first show that the phases of these sequences converge. As above, let us define α and β by the conditions a=|a|eiα, b=|b|eiβ, and |α-β|<π. Suppose that z and w are any two complex numbers such that z=|z|eiθ and w=|w|eiϕ with |ϕ-θ|<π. Then we have the following:

  • The phase of the geometric mean of z and w can be chosen to lie between θ and ϕ. This is because, as described earlier, this phase can be chosen as (θ+ϕ)/2.

  • The phase of the arithmetic mean of z and w can be chosen to lie between θ and ϕ.

By a simple induction argumentMathworldPlanetmath, these two facts imply that we can introduce polar representations an=|an|eiθn and gn=|gn|eiϕn where, for every n, we find that θn lies between α and β and likewise ϕn lies between α and β. Furthermore, since ϕn+1=(ϕn+θn)/2 and θn+1 lies between ϕn and θn, it follows that


Hence, we conclude that |ϕn-θn|0 as n. By the principle of nested intervals, we further conclude that the sequences {θn}n=0 and {ϕn}n=0 are both convergentMathworldPlanetmathPlanetmath and converge to the same limit.

Having shown that the phases converge, we now turn our attention to the moduli. Define mn=max(|an|,|gn|). Given any two complex numbers z,w, we have




so this sequence {mn}n=0 is decreasing. Since it bounded from below by 0, it converges.

Finally, we consider the ratios of the moduli of the arithmetic and geometric means. Define xn=|an|/|gn|. As in the real case, we shall derive a recursion relation for this quantity:

xn+1 =|an+1||gn+1|

For any real number x1, we have the following:

x-1 0
(x-1)2 0
x2-2x+1 0
x2+1 2x
x+1x 2

If 0<x<1, then 1/x>1, so we can swithch the roles of x and 1/x and conclude that, for all real x>0, we have


Applying this to the recursion we just derived and making use of the half-angle identity for the cosine, we see that

Title complex arithmetic-geometric mean
Canonical name ComplexArithmeticgeometricMean
Date of creation 2013-03-22 17:10:05
Last modified on 2013-03-22 17:10:05
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 15
Author rspuzio (6075)
Entry type Result
Classification msc 33E05
Classification msc 26E60