How Does Euler's Theorem Apply to Complex Eigenvalues in Differential Equations?

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In summary, the conversation is about working with complex eigenvalues in differential equations. The author only mentions Euler's Theorem and prefers to use a substitution when the roots are complex. The question is whether this substitution can still be called Euler's Theorem. The answer is that it can be called "Euler's theorem with -t substituted for t after applying the identity sin(-t)=-sin(t)". The person asking the question thanks the expert for clarifying and mentions that they were expecting some sort of identity for this substitution.
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FocusedWolf
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Ok, in my differential equation book, we're doing work on getting eigenvectors to complex eigenvalues.

anyway the author of the book only mentions Euler's Theorem as: http://rogercortesi.com/eqn/tempimagedir/eqn5095.png
and so he perfers to work with http://rogercortesi.com/eqn/tempimagedir/eqn7868.png when the roots are
eqn3362.png


So my question:

What is this called then?:
eqn9909.png

and can i use in place for
eqn4478.png
and still call it Euler's Theorem?
 
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  • #2
It's called "Euler's theorem with -t substituted for t after applying the identity sin(-t)=-sin(t)". Does it really need a name?
 
  • #3
Thx for that. I was sort of expecting some kind of identity to do that.
 
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1. What is Euler's Theorem?

Euler's Theorem is a mathematical theorem that states that for any positive integers a and m that are relatively prime, a to the power of phi(m) is congruent to 1 mod m, where phi(m) is the number of positive integers less than m that are relatively prime to m.

2. Who discovered Euler's Theorem?

Euler's Theorem was discovered by the Swiss mathematician Leonhard Euler in the 18th century.

3. What is the significance of Euler's Theorem?

Euler's Theorem has several applications in number theory and cryptography. It is also used to prove other important theorems, such as Fermat's Little Theorem.

4. Can Euler's Theorem be applied to non-integer values?

No, Euler's Theorem only applies to positive integers.

5. Is there a proof for Euler's Theorem?

Yes, there is a proof for Euler's Theorem, which involves concepts from number theory such as modular arithmetic and Euler's totient function. The proof can be found in most advanced textbooks on number theory.

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