find the principal value of log(1+i)
i don't understand the problem above, pls help, thanks.
Well, it's asking you to write this function in terms of its real and imaginary parts. Anyway, if , then that's the same as .
Equating real and imaginary parts gives and . Solving these simultaneously by dividing the first equation by the second gives
Substituting into the first equation gives
But since is real, that means everything inside the logarithm can only be positive, therefore we can only accept EVEN values of , so we'll write , which gives us and .
Therefore , and since this is multi-valued, we define the principal value as simply being the "first" possible value (in this case, where ).
So the principal value of .