Identity for exponential function with imaginary arguments

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SUMMARY

The identity Exp[iw/2] - Exp[-iw/2] = Exp[iw] - 1 is confirmed through simplification. The left-hand side simplifies to 2i sin(w/2), while the right-hand side simplifies to cos(w) + i sin(w) - 1. This establishes the identity as valid. The discussion emphasizes the importance of using exponential forms rather than complicating the proof with trigonometric identities.

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ParityCheck
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I have seen the following identity used.
Exp[iw/2]-Exp[iw/2]=Exp[iw]-1
I can't find this in any book and I can't prove it myself.
The left side equals 2isin(w/2)
The right side equals cos(w)+isin(w)-1
On the face of it, that seems to make the identity absurd
How can one go about proving such an identity? Or is it wrong?
 
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Welcome to PF!

ParityCheck said:
Exp[iw/2]-Exp[iw/2]=Exp[iw]-1
…
How can one go about proving such an identity? Or is it wrong?

Hi ParityCheck ! Welcome to PF! :smile:

(eiw/2 - e-iw/2)eiw/2 = eiw - 1 :wink:
 
Thanks. I was making it way too complicated with trig identities and series and...
 

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