What are the conditions for non-integer exponents to be single valued?

In summary, the Complex Number Paradox is a mathematical dilemma that arises when applying algebraic rules to complex numbers, resulting in seemingly contradictory solutions. This paradox is resolved by defining new rules for dealing with complex numbers, such as the use of complex conjugates and the imaginary unit, i. It has real-life applications in various fields, including signal processing and quantum mechanics. While initially counterintuitive, the paradox can be explained using geometric interpretations. There are ongoing debates and research efforts surrounding the best ways to teach and understand complex numbers.
  • #1
Nothing much, I have this:
gif.latex?\dpi{150}%20e^\theta=(e^{i\theta})^{-i}=(e^{i(\theta+2\pi)})^{-i}=e^{\theta+2\pi}.gif


I have studied (myself) about this for many days.
And I believe that, for some conditions
gif.latex?\dpi{150}%20(e^\psi)^\phi\neq%20e^{\psi\phi}.gif
for a complex ψ,ϕ

What are those conditions I mentioned about? And which field of study I should go to see?
 
Mathematics news on Phys.org
  • #2
Your first equation is obviously wrong, as you can see. For the second question, there are no conditions where it doesn't hold.

The problem lies in the fact that for non-integer exponents, the expression is not single valued.
 

1. What is the Complex Number Paradox?

The Complex Number Paradox is a mathematical dilemma in which the rules of algebra are applied to complex numbers, resulting in seemingly contradictory or illogical solutions.

2. How is the Complex Number Paradox solved?

The Complex Number Paradox is resolved by defining new rules for dealing with complex numbers, such as the concept of complex conjugates and the use of the imaginary unit, i. These rules allow for consistent and logical solutions to complex number equations.

3. What are some real-life applications of the Complex Number Paradox?

The Complex Number Paradox has applications in many areas of science and engineering, including signal processing, electrical engineering, and quantum mechanics. It is also used in the design of electronic circuits and in solving differential equations.

4. Can the Complex Number Paradox be explained intuitively?

While the rules for solving the Complex Number Paradox may seem counterintuitive at first, they can be explained using geometric interpretations and visual representations. With practice, these concepts can become more intuitive and easier to understand.

5. Are there any unresolved issues or controversies surrounding the Complex Number Paradox?

While the Complex Number Paradox has been extensively studied and resolved, there are still ongoing debates and discussions about the best ways to teach and understand complex numbers. There are also ongoing research efforts to extend the concept of complex numbers to higher dimensions.

Suggested for: What are the conditions for non-integer exponents to be single valued?

Replies
7
Views
1K
Replies
2
Views
1K
Replies
5
Views
929
Replies
8
Views
928
Replies
3
Views
681
Replies
7
Views
299
Replies
7
Views
1K
Replies
2
Views
718
Replies
5
Views
587
Back
Top