Complex Roots
What happens when the discriminant is negative? Sines and Cosines appear.
Introduction
If the roots of are complex, they always come in conjugate pairs:
If we use the formula from the previous section, we get . This is mathematically correct but involves complex numbers, which we want to avoid for physical models.
Euler's Formula
The Magic Identity
.
Using this, we can convert complex exponentials into real sines and cosines.
The general real-valued solution is:
- Exponential Part: controls the amplitude (growth/decay).
- Oscillatory Part: controls the frequency.
Visual: The Envelope
Interactive: Amplitude Modulation
If , the oscillations die out (Damped Vibration). If , they grow uncontrollably.
Worked Examples
Example 1: Basic Complex Roots
Solve .
1. Characteristic Equation:
.
2. Quadratic Formula:
.
.
So .
3. Solution:
.
Example 2: Pure Oscillation
Solve with .
1. Roots:
.
.
.
2. Apply :
.
3. Apply :
.
.
4. Final:
.
Example 3: Damped Motion
Solve using ?
1. Roots:
.
.
.
2. Apply BCs:
.
So .
(Always true for any ).
3. Conclusion:
There are infinitely many solutions of the form .
Practice Quiz
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