Question

What is the value of log(-1), in complex number?​

Answer 1

On the complex plane, ez=ez+2πkiez=ez+2πki for any integer kk. This is a bit funny, right? While the equality holds, we see that z≠z+2πkiz≠z+2πki ! So if we try the inverse of the exponential function, it's a bit ambiguous as to what number we get back. Does logez=zlog⁡ez=z or does logez=z+2πilog⁡ez=z+2πi?

The logarithm can return different complex numbers! It appears that the complex logarithm, is multi-valued. Here's a common definition:

logz=log|z|+iargz+i2πklog⁡z=log⁡|z|+iarg⁡z+i2πk

Where argzarg⁡z returns the argument or angle of the complex number zz. Note that the final term i2πki2πk accounts for the infinitely many values that can be returned. If we let z=reiϕz=reiϕ and substitute into the line above, we get

logz=logr+iϕ+i2πklog⁡z=log⁡r+iϕ+i2πk

Then

log(−1)=log|−1|+i(2k+1)π=i(2k

hope this will help you!!