San José State University 

appletmagic.com Thayer Watkins Silicon Valley & Tornado Alley USA 


Consider an expression of the form
where α is a complex number.
The infinitude may be expressed as the limit of an iteration of the form
Thus
The equation satisfied by ζ ;is
The way to find the complex root is to multiply the ratio 1/ζ top and bottom by the complex conjugate of ζ, ζ*. This results in
For example, suppose a value of α is sought such that the infinite exponentiation converges to 1+i. The complex conjugate of 1+i is 1i and 1+i²=2. We then want to find
The values of the iterations are
Iteration n  γ_{n} 
1  1.71896 + 0.38333i 
2  2.01543+1.3615i 
3  0.81326+2.1738i 
4  0.15864+0.9706i 
5  0.73751+0.48758i 
6  1.23539+0.57825i 
7  1.46449+0.99835i 
8  1.16337+1.4257i 
9  0.68803+1.23405i 
10  0.74362+0.84549i 
11  1.01367+0.75739i 
12  1.19562+0.91096i 
13  1.1476+1.13057i 
14  0.93839+1.16254i 
15  0.85597+1.00209i 
16  0.94889+0.89285i 
17  1.06344+0.92062i 
18  1.08697+1.02156i 
19  1.01229+1.07769i 
20  0.9427+1.03507i 
21  0.95365+0.96763i 
22  1.0086+0.95271i 
23  1.04031+0.98979i 
24  1.02223+1.02851i 
25  0.98498+1.02727i 
26  0.97359+0.99751i 
27  0.99278+0.97858i 
28  1.01424+0.9868i 
29  1.0154+1.00661i 
30  1.0001+1.01444i 
31  0.98869+1.00503i 
32  0.99216+0.99289i 
33  1.00284+0.99138i 
34  1.00777+0.99922i 
35  1.00331+1.00585i 
36  0.99654+1.00462i 
It does indeed appear that the iteration is converging to 1+i.
Now consider ζ=i. The complex conjugate of i is −i and ii*=1. Thus α would be i^{i}. Remarkably i^{i} is the real number 4.81048…. The exponential iteration of this number will not produce a complex number and will not converge.
A complex number may be expressed in three different forms. One is the rectangular form as x+iy, (where i is the square root of −1). A second is in polar form as Re^{iθ}. A third is the logarithmic form as e^{ρ+iθ}.
Arithmetic operations on two complex numbers can be simplified by choosing the proper forms for the two number. The addition α+β can be carried out most simply if both numbers are in the rectangular form; i.e., if α=x+iy and β=w+iz then α+β=(x+w)+i(y+z). Although multiplication of two complex numbers in rectangular forms is not very difficult multiplication is even easier if both are in polar form or both are in logarithmic form. If α=Re^{iθ} and β=re^{iφ} then α*β = (Rr)e^{i(θ+φ)}. If α=e^{ρ+iθ} and β=e^{σ+iφ} then α*β = e^{(ρ+σ)+i(θ+φ)}.
For some operations, such as exponentiation, the convenient forms may be different for α and β. For exponentiation let α=e^{ρ+iθ} and β=w+iz. Then
The polar form of ζ is Re^{i(ρz+θw)} where R=e^{ρw−θz}. The rectangular form is then Rcos(ρz+θw)+iRsin(ρz+θw).
By way of contrast let α=Re^{iθ} and β=x+iy. Then
This is the polar form of ζ. The rectangular form of ζ is
This works but clearly the logarithmic version is simpler.
Below is a calculator which implements complex exponentiation from the rectangular form of the complex numbers.
THW's
Complex Exponentiation c = a^b with a and b complex 


a  + i  
b  + i  
=  
Power c  + i  
Modulus  
Angle (deg) 
HOME PAGE OF Thayer Watkins 