How can complex numbers be elevated to complex powers?

• B
• ForceBoy
In summary, the conversation discusses a problem involving complex numbers raised to complex powers. It is found that some rules for exponentials that work for real numbers do not work for complex numbers. The conversation also explores different ways of representing complex numbers and the difficulties that arise when trying to raise them to complex powers. Ultimately, it is concluded that there is not a straightforward way to elevate numbers to complex powers and careful definitions and conventions are needed to understand the concept.
ForceBoy
Hello

I thought is would be fun to try a problem in which I had a complex number elevated to a complex power. To do this, I first tried to manipulate the general equation ## z^{w} ## (where ##z ## and ##w## are complex numbers) to look a bit more approachable. My work is as follows:

##z^{w}##
##z^{Re(w)+i Im(w)} ##
##z^{Re(w)}*e^{i Im(w) \ln(z)} ##

##\ln(z) = \ln|z| + \theta i ##

## z^{Re(w)}*e^{(\ln|z|+\theta i) i Im(w)} ##
## z^{Re(w)}*e^{i Im(w) \ln|z|- \theta Im(w)} ##
##z^{Re(w)}*e^{i Im(w) \ln|z|}*e^{-\theta Im(w)} ##
##z^{Re(w)} \frac{|z|^{i Im(w)}}{e^{\theta Im(w)}} ##

So
## z^{w} = z^{Re(w)} \frac{|z|^{i Im(w)}}{e^{\theta Im(w)}} ##

I now decided to test this equation out. I chose the number ## 1+ i ## and elevated it to the power of itself. Using wolframalpha I got that ## (1+i)^{(1+i)} \approx 0.273+ i0.584##
https://www.wolframalpha.com/input/?i=(1+i)^(1+i)

I then used the equation I ended up with and used the same number. In wolframalpha, ## (1+i)^{1} * \frac{\sqrt{2}^{i}}{e^{\frac{\pi}{2}}} \approx 0.125 + i0.266##.
https://www.wolframalpha.com/input/?i=(1+i)^1+((sqrt(2)^(i))/(e^(pi/2)))

I don't thing the values computed are even close enough to say that both expressions are equal so I must've messed up somewhere along the way. It would be very appreciated if someone could point out my mistake. In addition, are there any ways to elevate numbers to complex powers in a 'easier' way than just working from ##z^{w} ##? Thanks.

Some rules for exponentials that work for real numbers don't work for complex numbers. ln(x) is not unique any more, that causes problems as soon as you raise a complex number to a complex power.

Thanks. I understand better why my way doesn't work. Just to be clear,

##z^{Re(w)} * e^{(\ln|z| +\theta i)i Im(w)} ##
##z^{Re(w)} * e^{iIm(w)\ln(w)-\theta Im(w)} ##

Is this where I messed up? I think that here I used the rule ## (x^{a})^{b} = x^{ab} ##, where I multiplied the constants. The article said that this wasn't allowed for complex ##a## and ##b##. Also one more question, I arrived at the same answer as I had before when I used DeMoivre's Theorem

##z^{a} = |z|^{a}(\cos(a\theta)+i\sin(a\theta))##

The theorem only works for real exponents, right?

Your mistake seems to be using the wrong value of ##\theta##. The correct value is ##\pi/4##.

ForceBoy
D'oh! I just saw that! I can't believe I didn't see this before!

Now, I plugged the correct formula into wolfram alpha and got

## (1+i)^1*\frac{\sqrt{2}^{i}}{e^{\frac{\pi}{4}}} \approx 0.274 +i0.584 ##
https://www.wolframalpha.com/input/?i=(1+i)^1+((sqrt(2)^(i))/(e^(pi/4)))

This answer is now pretty close to the value of ##(1+i)^{(1+i)} ##. This but some of the operations I did might've not worked for complex numbers. What is 'the deal' then? Is ## 1+i ## an exception to the complex exponent 'rules' found on the article linked below or something else?

fresh_42 said:

ForceBoy said:
What is 'the deal' then?
If you want to be sure, then work with complex numbers in polar coordinates: ##z=r\cdot e^{i\varphi}##. Otherwise, you can also work with ##z=x+ t\cdot y \in \mathbb{R}[t]/(t^2+1)##. This might look complicated for complex powers, because it is! What is a polynomial up to another polynomial? It's explained in the insight article I linked.

fresh_42 said:
z=x+t⋅y∈R[t]/(t^{2}+1)
I'm sorry, I don't understand what the ##R[t]/(t^{2}+1)## means.

ForceBoy said:
I'm sorry, I don't understand what the ##R[t]/(t^{2}+1)## means.
It means that you can calculate with complex numbers as if they were real polynomials in one variable and always take the remainder of the division by ##t^2+1##. So e.g. ##t^3+ t^2 +2t +1 = (t+1)(t^2+1)+t \equiv t \operatorname{mod}(t^2+1)## would represent ##i##. This version of a representation of complex numbers should demonstrate where the problem is: we can easily multiply and add polynomials, but what should be ##t^t##? This requires some careful definitions and conventions how it should be understood, i.e. ##z^n## with ##z\in \mathbb{C}, n\in \mathbb{N}## is easy, ##z_1^{z_2}## is not.

I see what you're saying. I just can't understand why ##t^2 +1## should be used. I know it evaluates to zero if ##t=i## but I don't understand why ##t^2 +1 ## should be used.

I would also like to thank everyone at this point for your insight and help.

ForceBoy said:
I see what you're saying. I just can't understand why ##t^2 +1## should be used. I know it evaluates to zero if ##t=i## but I don't understand why ##t^2 +1 ## should be used.
This is the algebraic approach to the complex numbers. You identify all polynomials with zero, which have to be the zero as complex number. This means we start with all polynomials and identify all multiples of ##t^2+1## with ##0##. This is ##i^2+1=0## which is what defines complex numbers.

I'm starting to see what you mean. I'll try the polar approach then look at this algebraic approach better. Thanks.

1. What are complex numbers?

Complex numbers are numbers that consist of a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit (√-1).

2. How do you perform operations with complex numbers?

To add or subtract complex numbers, simply combine the real parts and the imaginary parts separately. To multiply complex numbers, use the FOIL method (First, Outer, Inner, Last) and remember that i² = -1. To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator.

3. What is the significance of the exponent in complex numbers?

The exponent in complex numbers represents repeated multiplication. For example, a complex number raised to the power of 2 means that the number is multiplied by itself. This is useful in simplifying complex expressions and solving equations involving complex numbers.

4. How do you convert a complex number from rectangular form to polar form?

To convert a complex number from rectangular form (a + bi) to polar form (r(cosθ + isinθ)), use the Pythagorean theorem to find the magnitude (r) and the inverse tangent function to find the angle (θ).

5. What are the properties of complex numbers?

Some properties of complex numbers include: commutative and associative properties for addition and multiplication, distributive property, and the existence of additive and multiplicative inverses. Additionally, the complex conjugate of a complex number has the same real part but the opposite sign for the imaginary part.

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