5. Complex Exponential Function

Try these first

1. 1.

   Show that e^(iπ) + 1 = 0 using Euler’s formula.

   Hint

   Think about the values of cosine and sine at π.

2. 2.

   Express cos(3x) using exponential functions.

   Hint

   Use Euler's formula.

3. 3.

   What does Euler's formula state?

   • e^(ix) = cos(x)
   • e^(ix) = cos(x) + i*sin(x)
   • e^(ix) = sin(x) + i*cos(x)

   Hint

   Remember, it connects e^(ix) with both cosine and sine.

4. 4.

   True or False: The modulus of e^(x + iy) is |e^x|.

   • True
   • False

   Hint

   Consider the definition of modulus for complex numbers.

5. 5.

   Prove that e^(iθ) has a modulus of 1 for any real θ. What does this imply about its representation on the complex plane?

   Hint

   Check the properties of sine and cosine for their squared sums.

6. 6.

   Use the properties of complex exponentials to solve for solutions to the equation d²y/dt² + 16y = 0, discussing its relationship to oscillatory motion.

   Hint

   Think about the connections between roots and oscillation frequency.