5.3. Properties of the Complex Exponential Function

Try these first

1. 1.

   What is the modulus of e^{2 + 3i}?

   Hint

   Remember the modulus only involves the real part.

2. 2.

   Write down Euler's formula.

   Hint

   Consider the relationship between exponential and trigonometric functions.

3. 3.

   What does the modulus of a complex exponential depend on?

   • Real part
   • Imaginary part
   • Both parts

   Hint

   Focus on the expression for modulus calculation.

4. 4.

   True or False: The derivative of the complex exponential function is equal to the function itself.

   • True
   • False

   Hint

   Think about how derivatives work for exponential functions in general.

5. 5.

   Prove that e^{(a+bi) + (c+di)} = e^{a+c} e^{(b+d)i} using the properties of complex exponentials.

   Hint

   Start by expanding both sides according to exponential properties.

6. 6.

   If |z| = e^3 and z has a negative imaginary part, what can we infer about its position in the complex plane?

   Hint

   Remember how modulus functions dictate positioning in the complex plane.