How to Calculate the Natural Logarithm of a Complex Number: ln(-1-i)

To find the natural logarithm of a complex number, we can use the formula:

ln(z) = ln|z| + i(arg(z) + 2πk)

where z is the complex number, ln|z| is the natural logarithm of the magnitude of z, arg(z) is the argument (or angle) of z, and k is any integer.

For the complex number -1 - i, the magnitude is √((-1)^2 + (-1)^2) = √(1 + 1) = √2.

The argument can be found using the arctan function:

arg(-1 - i) = arctan((-1)/(-1)) = arctan(1) = π/4.

Therefore, ln(-1 - i) = ln|√2| + i(π/4 + 2πk) = ln(√2) + i(π/4 + 2πk), where k is any integer.