Integrating complex exponentials with Infinite limits

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SUMMARY

The integration of the function exp(ix) - exp(-ix) with respect to x over infinite limits is feasible. This function can be expressed more succinctly as 2i sin(x) using Euler's formula. The integration can be performed along the real axis, and the anti-derivative can be verified for correctness. The discussion emphasizes the importance of understanding complex analysis when dealing with such integrals.

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  • Complex analysis fundamentals
  • Understanding of Euler's formula
  • Knowledge of integration techniques for complex functions
  • Familiarity with anti-derivatives
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  • Explore the application of Euler's formula in solving integrals
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Mathematicians, physics students, and anyone interested in advanced calculus and complex analysis will benefit from this discussion.

Ribena
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Hi all,

I'm trying to integrate the function below with respect to x

exp(ix)-exp(-ix)

With infinity and negative infinity as the limits. Would the integration be possible?
 
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There is a shorter way to express this function.
Alternatively, you can find the anti-derivative, and check yourself.

As you are working with complex numbers, you should add that you try to integrate this along the real axis (well, I would expect this).

Would the integration be possible?
What do you think, and why?
 

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