Complex exponential to a power

You'll learn much more than what is needed for this particular situation.In summary, the conversation discusses the application of Euler's identity to rational numbers, and how it differs from its application to integers. The use of real number rules on complex numbers is also explored, and it is suggested to study the complex exponential function in the context of complex analysis for a better understanding.
  • #1
Mr Davis 97
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Say I have ##e^{2\pi i n}##, where ##n## is an integer. Then it's clear that ##(e^{2\pi i})^n = 1^n = 1##.
However, what if replace ##n## with a rational number ##r##? It seems that by the same reasoning we should have that ##e^{2\pi i r} = (e^{2\pi i})^r = 1^r = 1##. But what if ##r=1/2## for example? Then ##e^{2 \pi i (\frac{1}{2})} = e^{\pi i} = -1##. What am I doing wrong here?
 
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  • #2
Isn't that Euler's identity you now have?

Add 1 to both sides to get ##e^{i\pi} + 1 = -1 + 1## hence ##e^{i\pi} + 1 = 0 ##

So sqrt of 1 has two roots 1 and -1 right? squaring the answer you got brings you back to the original ##e^{2\pi i} = 1##

@fresh_42 or @Mark44 can provide a better answer I think.
 
  • #5
Mr Davis 97 said:
Say I have ##e^{2\pi i n}##, where ##n## is an integer. Then it's clear that ##(e^{2\pi i})^n = 1^n = 1##.
However, what if replace ##n## with a rational number ##r##? It seems that by the same reasoning we should have that ##e^{2\pi i r} = (e^{2\pi i})^r = 1^r = 1##.
No more than the fact that ##sin(2\pi n) == 0## for all ##n\in N## implies ##sin(2\pi r) == 0## for all rational numbers ##r##
But what if ##r=1/2## for example? Then ##e^{2 \pi i (\frac{1}{2})} = e^{\pi i} = -1##. What am I doing wrong here?
The complex exponential function ##e^{i\theta}, \theta \in R## is very fundamental. I suggest that you study it in the context of complex analysis.
 

1. What is a complex exponential to a power?

A complex exponential to a power is a mathematical expression of the form eix, where e is the base of natural logarithms and x is a complex number. It is also known as a complex exponential function.

2. How is a complex exponential to a power calculated?

A complex exponential to a power can be calculated using the Euler's formula: eix = cos(x) + i*sin(x). This formula relates the complex exponential to trigonometric functions and allows for easier computation.

3. What are the properties of a complex exponential to a power?

Some key properties of complex exponential to a power include:

  • The values of the function repeat every 2π radians or 360 degrees
  • The derivative of the function is itself, making it a unique function
  • The function is periodic and has no real roots

4. Why is complex exponential to a power important in mathematics?

Complex exponential to a power has many applications in mathematics and other fields such as physics, engineering, and signal processing. It is useful for modeling periodic phenomena, analyzing oscillatory systems, and solving differential equations.

5. Can a complex exponential to a power have a negative exponent?

Yes, a complex exponential to a power can have a negative exponent. This results in a reciprocal of the positive exponent, which can be expressed as e-ix = 1/eix = 1/(cos(x) + i*sin(x)). This can be useful in simplifying complex expressions or solving certain problems in mathematics.

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