Is Euler's Identity Applicable to Transforming f(x)=constant*e^(-x^2)?

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Discussion Overview

The discussion revolves around the applicability of Euler's identity to the function f(x) = constant * e^(-x^2). Participants explore whether this function can be rewritten in a form that allows the use of Euler's identity, particularly considering its representation in terms of cosine and sine.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant questions if f(x) can be rewritten as f(x) = constant * e^(ix)^2 to utilize Euler's identity.
  • Another participant points out that the expression exp((ix)^2) is not equivalent to (exp(ix))^2, indicating a misunderstanding in the manipulation of the exponential function.
  • It is suggested that while one can express exp(-x^2) as exp(i * (ix^2)), this leads to imaginary arguments for sine and cosine, which may not be useful.
  • A further comment mentions an equivalent formula involving sinh and cosh, although its utility is questioned.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the applicability of Euler's identity to the given function, with multiple viewpoints and some confusion regarding the mathematical manipulations involved.

Contextual Notes

There are limitations in the understanding of the exponential function's properties and the implications of using imaginary arguments in the context of Euler's identity.

zpatenaude37
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I have a homework question and I am wondering if you can use Eulers identity in this case.

If the equation is f(x)=constant*e^(-x^2) can this be rewritten as f(x)=consant*e^(ix)^2
and then, can you use the identity when it is in this form?

Edit: Can it be put in the form cosx+isinx

I am not well acquainted with Eulers Identity so bear with me
 
Last edited:
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zpatenaude37 said:
I have a homework question and I am wondering if you can use Eulers identity in this case.

If the equation is f(x)=constant*e^(-x^2) can this be rewritten as f(x)=consant*e^(ix)^2
and then, can you use the identity when it is in this form?

I am not well acquainted with Eulers Identity so bear with me
What do you propose to do with $${e^{(ix)}}^2$$
 
sorry edited for clarity
 
zpatenaude37 said:
sorry edited for clarity
Sadly ##exp((ix)^2) \ne (exp(ix))^2## if that's what you intended.
 
You can use ##\exp(-x^2) = \exp(i (ix^2))## and use the Euler formula for that expression, but that gives imaginary arguments for the sine and cosine, which does not look helpful.
 
zpatenaude37 said:
I have a homework question and I am wondering if you can use Eulers identity in this case.

If the equation is f(x)=constant*e^(-x^2) can this be rewritten as f(x)=consant*e^(ix)^2
and then, can you use the identity when it is in this form?

Edit: Can it be put in the form cosx+isinx

I am not well acquainted with Eulers Identity so bear with me
There is an equivalent formula involving sinh and cosh, but I doubt if it would help you.
 

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