How to calculate the natural logarithm of a complex number?

The natural logarithm of a complex number can be calculated using the formula ln(z) = ln|z| + i arg(z), where z is a complex number.

To calculate the natural logarithm of a complex number, we first need to find its modulus and argument. Let z = a + bi be a complex number, where a and b are real numbers and i is the imaginary unit. The modulus of z is given by |z| = √(a^2 + b^2), and the argument of z is given by arg(z) = tan^-1(b/a) if a > 0, or arg(z) = tan^-1(b/a) + π if a < 0. For more on the types and properties of numbers, see Types of Numbers.

Once we have found the modulus and argument of z, we can use the formula ln(z) = ln|z| + i arg(z) to calculate its natural logarithm. For example, let z = 2 + 3i be a complex number. Then |z| = √(2^2 + 3^2) = √13, and arg(z) = tan^-1(3/2) = 1.249 radians (since a > 0). Therefore, ln(z) = ln|z| + i arg(z) = ln(√13) + i(1.249) ≈ 1.322 + 1.249i. Understanding the trigonometric form of complex numbers can further enhance this calculation; more details can be found on Trigonometric Form of Complex Numbers.

It is important to note that the natural logarithm of a complex number is not unique, since the argument of a complex number is only defined up to an integer multiple of 2π. Therefore, ln(z) = ln|z| + i(arg(z) + 2πn), where n is an integer, is also a valid natural logarithm of z. Additional insights into the argument calculations can be accessed through Inverse Trigonometric Functions.

A-Level Maths Tutor Summary: To calculate the natural logarithm of a complex number, first find its modulus (|z| = √(a^2 + b^2)) and argument (arg(z), which depends on the values of a and b). Use these to apply the formula ln(z) = ln|z| + i arg(z). Remember, the result isn't unique due to the argument's nature, meaning we can add 2πn (where n is any integer) to the argument.