Simple math Question concerning the natural logarithm of Complex Number

Question

There is this simple exercise, in which the complex number is given in polar form as z= mod=|10|,arg=322.75 degrees and i must find the ln of it. So to do that i must first convert the complex number into exponential form and then proceed to find the natural logarithm.

It is an exercise from a book i use, but i was confused and unsure about one thing. The book is giving the answer which is ln(z)=2.303 -j*0.65. First he converted 322.75 degrees into radians and then it found the natural logarithm as usual.. But he chose to convert it first to -360+322.75=-37.25 degrees and then he converted to radians which is -0.65.

What i did though is to convert 322.75 degrees to radians from the start (which is 5.633) and then i found the natural logarithm of the complex number and what i found is ln(z) = 2.303 +j*5.633.

This answer differs from the one on the book and i just need to verify whether my answer is correct. I know that 322.75 degrees is the same with -37.25 degrees, so my answer must be correct as well (i think). But i would like to make sure. So, if anyone could help me on that, it would help me very much. Thanks a lot in advance.

Answer 1

Both answers are correct and you actually put your finger on a deep mathematical concept: the fact that the complex logarithm has several branches, all “spiraling” around the zero point. For more explanations, see the Wikipedia page (for a start).

However, since this a math stackexchange, I point out that it would be better advised to give an exact answer first, before giving an approximate one. Since $322.75 = \frac{1291}{4}$, your angle is $\frac{1291}{720} \pi$ radians, so that the logarithm of your number is $$ \log 10 + \frac{1291}{720} i \pi \pmod{2 i \pi}.$$ (That's assuming that the 322.75 value you gave is correct. With 322.45 it is a slightly different shade of ugly).

This is the value you computed. For the book answer, just subtract $2 i \pi$ to obtain $\log 10 - \frac{149}{720} i \pi$. The reason to use either answer is linked to the choice of the determination of the logarithm. It is usual to use the determination with the smallest absolute value for the imaginary part, thus the answer of the book.

Answer 2

Both answers are correct. The log of any nonzero complex number has an infinite number of distinct values, each of which differs by an integral multiple of $2\pi i$.

The author of the problem may be interested in values of the argument that lie in $(-\pi,\pi]$.