San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 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

Thus

#### ζ = limn→∞ ζn

The equation satisfied by ζ ;is

#### ζ = αζand 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

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=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

 Iterationn γn 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. 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 = (Reiθ)x+iy = Rx+iyeiθ(x+iy) = RxRiyeiθx−θy = Rxeiyln(R)eiθxe−θywhich 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)