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What is the Laplace Transform of $1$?
What is the Laplace Transform of $1$?
What is the Laplace Transform of $e^{at}$?
What is the Laplace Transform of $e^{at}$?
What is the Laplace Transform of $t^n$ where $n = 1, 2, 3,...$?
What is the Laplace Transform of $t^n$ where $n = 1, 2, 3,...$?
What is the Laplace Transform of $t^p$ where $p > -1$?
What is the Laplace Transform of $t^p$ where $p > -1$?
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What is the Laplace Transform of $\sin(at)$?
What is the Laplace Transform of $\sin(at)$?
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What is the Laplace Transform of $\cos(at)$?
What is the Laplace Transform of $\cos(at)$?
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What is the Laplace Transform of $t \sin(at)$?
What is the Laplace Transform of $t \sin(at)$?
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The Laplace Transform of $\delta(t - c)$ is $e^{-cs}$.
The Laplace Transform of $\delta(t - c)$ is $e^{-cs}$.
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What is the Laplace Transform of $e^{at} \cos(bt)$?
What is the Laplace Transform of $e^{at} \cos(bt)$?
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The Laplace Transform of $t^n e^{at}$ is ___ when $n = 1, 2, 3,...$.
The Laplace Transform of $t^n e^{at}$ is ___ when $n = 1, 2, 3,...$.
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Study Notes
Laplace Transforms
- The Laplace Transform is a mathematical tool used to transform a function of time, f(t), to a function of complex frequency, F(s).
- This transformation simplifies solving differential equations, especially those with discontinuous inputs like impulse functions.
- F(s) is the Laplace Transform of f(t) and is denoted by L{f(t)}.
- f(t) is the inverse Laplace Transform of F(s) and is denoted by L−1{F(s)}.
Laplace Transform Pairs
- 1. Unit Step Function: The transform of the unit step function (1) is 1/s.
- 2. Exponential Function: The transform of e^(at) is 1/(s-a).
- 3. Power Function: The transform of t^n is n! / s^(n+1).
- 4. Fractional Power Function: The transform of t^p (where p>-1) is Γ(p+1)/s^(p+1).
- 5. Square Root of Time: The transform of √t is √(π) / (2s^(3/2)).
- 6. Odd Powers of Time: The transform of t^(n-1/2) (where n is a positive integer) is (135*…(2n-1))√(π) / (2^n * s^(n+1/2)).
- 7. Sine Function: The transform of sin(at) is a / (s^2 + a^2).
- 8. Cosine Function: The transform of cos(at) is s / (s^2 + a^2).
- 9. Time Multiplied by Sine: The transform of t*sin(at) is 2as / (s^2 + a^2)^2.
- 10. Time Multiplied by Cosine: The transform of t*cos(at) is (s^2 - a^2) / (s^2 + a^2)^2.
- 11. Modified Sine: The transform of sin(at) - at*cos(at) is 2a^3 / (s^2 + a^2)^2.
- 12. Modified Cosine: The transform of sin(at) + at*cos(at) is 2as^2 / (s^2 + a^2)^2.
- 13. Modified Cosine: The transform of cos(at) - at*sin(at) is s(s^2 - a^2) / (s^2 + a^2)^2.
- 14. Modified Cosine: The transform of cos(at) + at*sin(at) is s(s^2 + 3a^2) / (s^2 + a^2)^2.
- 15. Shifted Sine: The transform of sin(at + b) is (ssin(b) + acos(b)) / (s^2 + a^2).
- 16. Shifted Cosine: The transform of cos(at + b) is (scos(b) - asin(b)) / (s^2 + a^2).
- 17. Hyperbolic Sine: The transform of sinh(at) is a / (s^2 - a^2).
- 18. Hyperbolic Cosine: The transform of cosh(at) is s / (s^2 - a^2).
- 19. Exponential Sine: The transform of e^(at)*sin(bt) is b / ((s-a)^2 + b^2).
- 20. Exponential Cosine: The transform of e^(at)*cos(bt) is (s-a) / ((s-a)^2 + b^2).
- 21. Exponential Hyperbolic Sine: The transform of e^(at)*sinh(bt) is b / ((s-a)^2 - b^2).
- 22. Exponential Hyperbolic Cosine: The transform of e^(at)*cosh(bt) is (s-a) / ((s-a)^2 - b^2).
- 23. Exponential Power Function: The transform of t^n*e^(at) (where n is a positive integer) is n! / (s-a)^(n+1).
- 24. Scaled Function: The transform of f(ct) is (1/c)*F(s/c).
- 25. Shifted Unit Step: The transform of u(t-c), the unit step function shifted by c, is e^(-cs)/s.
- 26. Dirac Delta Function: The transform of δ(t-c), the Dirac delta function at t=c, is e^(-cs).
Important Notes
- Γ(p): The Gamma function is a generalization of the factorial function for complex numbers.
- u(t-c): This represents the unit step function, which is 0 for t < c and 1 for t ≥ c.
- δ(t-c): This represents the Dirac delta function, which is 0 for t ≠ c and has an infinite value at t = c.
- c: This represents a constant value that can be interpreted as a shift or scaling factor within the function.
- s: This represents the complex frequency variable in the Laplace transform domain.
These Laplace transform pairs are useful for solving a wide range of problems in engineering and physics, especially in areas like circuit analysis, control systems, and vibration analysis.
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Description
This quiz explores the concepts of Laplace Transforms, including their definitions and how they simplify solving differential equations. You will also learn about common Laplace Transform pairs, providing a foundation for further studies in engineering and mathematics.
