Proof of i-th Root of i: Analytic Connection?

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In summary: It's funny because it's true.In summary, the proof is algebraic and there is no evidence it will work anywhere else.
  • #1
Gear300
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TL;DR Summary
imaginary algebra
I saw a proof in which they came up with the ith root of i through the typical algebra.
$$
i^{1/i} = i^{-i} = e^{i\frac{\pi}{2} \cdot -i} = e^{\frac{\pi}{2}} ~.
$$
But it seems the proof is entirely algebraic, so we have no grounds for thinking it works anywhere. The only exception might be an analytic connection with power series, like a Laurent series. Is there such a connection, or is this as ad hoc as it seems?
 
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  • #2
Gear300 said:
TL;DR Summary: imaginary algebra

I saw a proof in which they came up with the ith root of i through the typical algebra.
$$
i^{1/i} = i^{-i} = e^{i\frac{\pi}{2} \cdot -i} = e^{\frac{\pi}{2}} ~.
$$
But it seems the proof is entirely algebraic, so we have no grounds for thinking it works anywhere. The only exception might be an analytic connection with power series, like a Laurent series. Is there such a connection, or is this as ad hoc as it seems?
It's pretty straightforward.
##i = e^{i\pi/2} \Rightarrow i^{-i} = (e^{i\pi/2})^{-i} = e^{(i\pi/2) \cdot (-i)} = e^{\pi/2}##
 
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  • #3
Mark44 said:
It's pretty straightforward.
##i = e^{i\pi/2} \Rightarrow i^{-i} = (e^{i\pi/2})^{-i} = e^{(i\pi/2) \cdot (-i)} = e^{\pi/2}##
So it's fine doing this sort of thing in a quantum mechanics equation?
 
  • #4
Gear300 said:
TL;DR Summary: imaginary algebra

I saw a proof in which they came up with the ith root of i through the typical algebra.
$$
i^{1/i} = i^{-i} = e^{i\frac{\pi}{2} \cdot -i} = e^{\frac{\pi}{2}} ~.
$$
But it seems the proof is entirely algebraic, so we have no grounds for thinking it works anywhere. The only exception might be an analytic connection with power series, like a Laurent series. Is there such a connection, or is this as ad hoc as it seems?
Of course, this isn't unique.
##i^{1/i} = \left ( e^{i \pi / 2} \right )^{1/i} = \left ( e^{i \pi / 2 + 2 k \pi i } \right )^{1/i} = e^{\pi /2 + 2 k \pi}##
where k is any integer, so

##i^{1/i} = e^{5 \pi / 2}##
just as well.

-Dan
 
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  • #5
Gear300 said:
So it's fine doing this sort of thing in a quantum mechanics equation?
Why not? This is mathematics.
 
  • #6
topsquark said:
Of course, this isn't unique.
##i^{1/i} = \left ( e^{i \pi / 2} \right )^{1/i} = \left ( e^{i \pi / 2 + 2 k \pi i } \right )^{1/i} = e^{\pi /2 + 2 k \pi}##
where k is any integer, so

##i^{1/i} = e^{5 \pi / 2}##
just as well.

-Dan

Mark44 said:
Why not? This is mathematics.
True enough. I guess it doesn't work unless it works.
 
  • #7
Far from "ad hoc"; these are basic properties of exponents and polar coordinates are the best way to deal with powers in the complex plane.
I admit that I have no geometric image of numbers to complex powers, but I have to accept it because so much of it works out perfectly for real powers.
 
  • #8
Gear300 said:
True enough. I guess it doesn't work unless it works.
I know that ##i ^i \approx 0.2##, which is quite funny.
 
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FAQ: Proof of i-th Root of i: Analytic Connection?

1. What is the proof of the i-th root of i?

The proof of the i-th root of i involves using complex numbers and the fundamental theorem of algebra to show that i raised to the power of 1/i is equal to i.

2. Why is the i-th root of i considered an analytic connection?

The i-th root of i is considered an analytic connection because it is a mathematical relationship between two complex numbers that can be expressed using analytic functions, which are functions that can be represented by a power series.

3. How is the proof of the i-th root of i relevant in mathematics?

The proof of the i-th root of i is relevant in mathematics because it helps to understand the properties of complex numbers and their relationships, which are important in various fields such as engineering, physics, and computer science.

4. Are there any real-world applications of the i-th root of i?

Yes, the i-th root of i has applications in signal processing, electrical engineering, and quantum mechanics. It is also used in the study of fractals and chaos theory.

5. What are some other interesting mathematical connections involving the i-th root of i?

One interesting connection is the relationship between the i-th root of i and the golden ratio, which is a mathematical constant found in nature and art. The i-th root of i is also connected to the Mandelbrot set, a famous fractal pattern in complex dynamics.

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