complex arithmetic-geometric mean
It is also possible to define the arithmetic-geometric mean for complex numbers. To do this, we first must make the geometric mean unambiguous by choosing a branch of the square root. We may do this as follows: Let and br two non-zero complex numbers such that for any real number . Then we will say that is the geometric mean of and if and is a convex combination of and (i.e. for positive real numbers and ).
Geometrically, this may be understood as follows: The condition means that the angle between and differs from . The square root of will lie on a line bisecting this angle, at a distance from . Our condition states that we should choose such that bisects the angle smaller than , as in the figure below:
Analytically, if we pick a polar representation , with , then . Having clarified this preliminary item, we now proceed to the main definition.
We shall first show that the phases of these sequences converge. As above, let us define and by the conditions , , and . Suppose that and are any two complex numbers such that and with . Then we have the following:
The phase of the geometric mean of and can be chosen to lie between and . This is because, as described earlier, this phase can be chosen as .
The phase of the arithmetic mean of and can be chosen to lie between and .
By a simple induction argument, these two facts imply that we can introduce polar representations and where, for every , we find that lies between and and likewise lies between and . Furthermore, since and lies between and , it follows that
Hence, we conclude that as . By the principle of nested intervals, we further conclude that the sequences and are both convergent and converge to the same limit.
Having shown that the phases converge, we now turn our attention to the moduli. Define . Given any two complex numbers , we have
Finally, we consider the ratios of the moduli of the arithmetic and geometric means. Define . As in the real case, we shall derive a recursion relation for this quantity:
For any real number , we have the following:
If , then , so we can swithch the roles of and and conclude that, for all real , 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|
|Date of creation||2013-03-22 17:10:05|
|Last modified on||2013-03-22 17:10:05|
|Last modified by||rspuzio (6075)|