5.11. Logarithm of a Complex Number

Try these first

1. 1.

   Write the logarithm of the complex number z = 3e^(iπ/4) in its general form.

   Hint

   Remember to express it in terms of modulus and argument!

2. 2.

   What is the principal value of the logarithm for z = 2e^(iπ)?

   Hint

   Consider the angle for the principal value!

3. 3.

   What is the formula for the complex logarithm?

   Hint

   Think of the polar form of a complex number.

4. 4.

   True or False: The complex logarithm is a single-valued function.

   Hint

   Remember the periodic properties of complex exponentials.

5. 5.

   Prove that ln(z) = ln(r) + i(θ + 2πn) using the periodic property of complex exponentials.

   Hint

   Start with the definition of the complex exponential.

6. 6.

   Derive the principal value for ln(-1) and discuss its significance in engineering applications.

   Hint

   Consider the angle that corresponds to -1 in the complex plane.