|San José State University|
& Tornado Alley
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
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 1-i and |1+i|=2. We then want to find
The values of the iterations are
|1||1.71896 + 0.38333i|
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.
For an illustration of why there may be no convergence see Infinite Exponentiation.
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 Reiθ. 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=Reiθ and β=reiφ 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
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=Reiθ 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.
c = a^b with a and b complex
|Power c||+ i|
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