How to calculate the logarithm of a complex number?

To calculate the logarithm of a complex number, use the formula log(z) = ln|z| + i arg(z).

The logarithm of a complex number is defined as the power to which e must be raised to obtain the complex number. It is not a single value, but rather a set of values that differ by multiples of 2πi. Therefore, we must specify a branch of the logarithm function.

To calculate the logarithm of a complex number z = x + iy, we first find its modulus |z| = √(x^2 + y^2) and argument arg(z) = tan^-1(y/x) (taking care to adjust for the correct quadrant).

Then, using the formula log(z) = ln|z| + i arg(z), we can find the logarithm of z. For example, let z = 2 + 3i.

|z| = √(2^2 + 3^2) = √13
arg(z) = tan^-1(3/2) = 1.249 radians

Therefore, log(z) = ln|z| + i arg(z) = ln(√13) + i(1.249) ≈ 1.322 + 1.249i.

Note that this is just one possible value of the logarithm of z. To find other values, we can add multiples of 2πi to the imaginary part. For example, log(z) = 1.322 + 1.249i + 2πni, where n is an integer.