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Infinite Complex Exponentiation

Consider an expression of the form

ζ = ααα

where α is a complex number.

The infinitude may be expressed as the limit of an iteration of the form

ζn = αζn-1

Thus

ζ = limn→∞ ζn

The equation satisfied by ζ ;is

ζ = αζ
and thus α = ζ1/ζ

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

α = ζζ*/|ζ|²

There May Be a Finite Limit for Infinite Complex Exponentiation

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 1-i and |1+i|=2. We then want to find

(1+i)1/(1+i) = (1+i)(1-i)/2 = 1.71896 + 0.38333i

The values of the iterations are

Iteration
n
γn
11.71896 + 0.38333i
22.01543+1.3615i
30.81326+2.1738i
40.15864+0.9706i
50.73751+0.48758i
61.23539+0.57825i
71.46449+0.99835i
81.16337+1.4257i
90.68803+1.23405i
100.74362+0.84549i
111.01367+0.75739i
121.19562+0.91096i
131.1476+1.13057i
140.93839+1.16254i
150.85597+1.00209i
160.94889+0.89285i
171.06344+0.92062i
181.08697+1.02156i
191.01229+1.07769i
200.9427+1.03507i
210.95365+0.96763i
221.0086+0.95271i
231.04031+0.98979i
241.02223+1.02851i
250.98498+1.02727i
260.97359+0.99751i
270.99278+0.97858i
281.01424+0.9868i
291.0154+1.00661i
301.0001+1.01444i
310.98869+1.00503i
320.99216+0.99289i
331.00284+0.99138i
341.00777+0.99922i
351.00331+1.00585i
360.99654+1.00462i

It does indeed appear that the iteration is converging to 1+i.

There May Be No Finite Limit for Infinite Complex Exponentiation

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.

For an illustration of why there may be no convergence see Infinite Exponentiation.


The Different Forms for Expressing a Complex Number

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. A third is the logarithmic form as eρ+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 and β=re then α*β = (Rr)ei(θ+φ). If α=eρ+iθ and β=eσ+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=eρ+iθ and β=w+iz. Then

ζ = αβ = (eρ+iθ)w+iz = e(ρ+iθ)(w+iz)
which reduces to
ζ = e(ρw−θz)+i(ρz+θw)
 

The polar form of ζ is Rei(ρz+θw) where R=eρw−θz. The rectangular form is then Rcos(ρz+θw)+iRsin(ρz+θw).

By way of contrast let α=Re and β=x+iy. Then

ζ = (Re)x+iy = Rx+iyeiθ(x+iy)
= RxRiyeiθx−θy = Rxeiyln(R)eiθxe−θy
which upon combining the like terms gives
=[Rxe−θy] ei(yln(R)+xθ)
or, equivalently

=[exln(R)−θy] ei(yln(R)+xθ)

This is the polar form of ζ. The rectangular form of ζ is

eln(R)x−θycos(yln(R)+xθ) + ieln(R)x−θysin(yln(R)+xθ)

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)


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