ln(i) using Euler's Equation

Problem: Using Euler's Equation, evaluate \(\ln i\). Euler's Equation is given by: \[ e^{ix}=\cos x +i\sin x\] (a proof of this equation can be found here)

 

Solution:

Consider what we get if we \(ln\) both sides of Euler's Equation:

\[ \ln\left(e^{ix}\right)=\ln(\cos x +i\sin x)\]

\[ ix=\ln(\cos x +i\sin x)\]

 

Now we want \(\ln i\) , so notice that we can get that on the RHS if we let \(x=\frac{\pi}{2}\):

\[ i\frac{\pi}{2}=\ln \left(\cos \frac{\pi}{2} +i\sin \frac{\pi}{2}\right)\]

\[ i\frac{\pi}{2}=\ln(i)\]

  

\[\therefore \ln i = \frac{\pi i}{2}\]

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