Laplace Transform of te^(-at)u(t)
The Laplace transform is a powerful tool for analyzing linear systems and differential equations. For the function te^(-at)u(t), where u(t) is the unit step function, we can derive its Laplace transform using integration.
Step 1: Definition
The Laplace transform L{f(t)} is defined as:
- L{f(t)} = ∫[0 to ∞] e^(-st) f(t) dt
Step 2: Applying the Definition
For f(t) = te^(-at)u(t):
- L{te^(-at)} = ∫[0 to ∞] e^(-st) te^(-at) dt
- = ∫[0 to ∞] te^(-(s + a)t) dt
Step 3: Solving the Integral
Using the formula for the Laplace transform of t^n:
- L{t^n} = n!/s^(n + 1)
In this case, n = 1:
- L{te^(-at)} = 1! / (s + a)^(2)
Thus, the final result is:
- L{te^(-at)u(t)} = 1 / (s + a)^(2)
Laplace Transform of cos(ωt)u(t)
The cosine function is also commonly analyzed using the Laplace transform.
Step 1: Definition
Again, using the definition:
- L{f(t)} = ∫[0 to ∞] e^(-st) f(t) dt
Step 2: Applying the Definition
For f(t) = cos(ωt)u(t):
- L{cos(ωt)} = ∫[0 to ∞] e^(-st) cos(ωt) dt
Step 3: Solving the Integral
The formula for the Laplace transform of cos(ωt) is:
- L{cos(ωt)} = s / (s^2 + ω^2)
Thus, the final result is:
- L{cos(ωt)u(t)} = s / (s^2 + ω^2)
Conclusion
The Laplace transforms are fundamental in system analysis, providing insights into system behavior in the frequency domain.