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HuskyLab
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I'm trying to get a more intuitive understanding of Euler's identity, more specifically, what raising e to the power of i means and why additionally raising by an angle in radians rotates the real value into the imaginary plane. I understand you can derive Euler's formula from the cosx, sinx and ex Taylor series with the addition of i to form the identity. I understand the algebra but is this property only exclusive to e^x ? Does raising let's say 2^{i*pi} mean anything at all? The only thing we are changing is the base, after all e is just a constant. I had a quick look at the Taylor series for 2^x but by a quick comparison, there were some nasty constants, no direct relation seemed apparent. If I said anything that's wrong just me know. Thanks.