Laplace Transform Cheat Sheet: Essential Formulas and Table

The Laplace Transform is a powerful tool to simplify solving differential equations by converting them into algebraic ones. This cheat sheet provides essential formulas, properties, and common transforms, making tackling problems in control systems, signal processing, and engineering analysis easier.

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1. Definition:

Laplace transform of a function $f(t)$ is:

$F(s) = \int_0^{+\infty} f(t).e^{-st} . dt$

2. Laplace Transform Formulas:

Given $f(t) = \mathcal{L}^{-1}\{F(s)\}$ then $F(s) = \mathcal{L}\{f(t)\}$
Below is the table for Laplace transform for all types of possible functions.
$\space \space \space \bold{f(t)} \space \space \space \space \space \space \space \space \space \space \space \space \bold{F(s)}$

1. $1 \quad \Rightarrow \quad \dfrac{1}{s}$

   
   

2. $e^{at} \quad \Rightarrow \quad \dfrac{1}{s - a}$

   
   

3. $t^n, \quad n = 1, 2, 3, \dots \quad \Rightarrow \quad \dfrac{n!}{s^{n+1}}$

   
   

4. $t^p, \quad p > -1 \quad \Rightarrow \quad \dfrac{\Gamma(p+1)}{s^{p+1}}$

   
   

5. $\sqrt{t} \quad \Rightarrow \quad \dfrac{\sqrt{\pi}}{2s^{(\dfrac{3}{2})}}$

   
   

6. $t^{(n-\dfrac{1}{2})}, \quad n = 1, 2, 3, \dots \quad \Rightarrow \quad \dfrac{1 \cdot 3 \cdot 5 \cdot \dots \cdot (2n - 1)\sqrt{\pi}}{2^n s^{(n + \dfrac{1}{2})}}$

   
   

7. $\sin(at) \quad \Rightarrow \quad \dfrac{a}{s^2 + a^2}$

   
   

8. $\cos(at) \quad \Rightarrow \quad \dfrac{s}{s^2 + a^2}$

   
   

9. $t \space \sin(at) \quad \Rightarrow \quad \dfrac{2as}{(s^2 + a^2)^2}$

   
   

10. $t \space \cos(at) \quad \Rightarrow \quad \dfrac{s^2 - a^2}{(s^2 + a^2)^2}$

    
    

11. $\sin(at) - at \space \cos(at) \quad \Rightarrow \quad \dfrac{2a^3}{(s^2 + a^2)^2}$

    
    

12. $\sin(at) + at \space \cos(at) \quad \Rightarrow \quad \dfrac{2as^2}{(s^2 + a^2)^2}$

    
    

13. $\cos(at) - at \space \sin(at) \quad \Rightarrow \quad \frac{s(s^2 - a^2)}{(s^2 + a^2)^2}$

    
    

14. $\cos(at) + at \space \sin(at) \quad \Rightarrow \quad \dfrac{s(s^2 + 3a^2)}{(s^2 + a^2)^2}$

    
    

15. $\sin(at + b) \quad \Rightarrow \quad \dfrac{s \space \sin(b) + a \space \cos(b)}{s^2 + a^2}$

    
    

16. $\cos(at + b) \quad \Rightarrow \quad \dfrac{s \space \cos(b) - a \space \sin(b)}{s^2 + a^2}$

    
    

17. $\sinh(at) \quad \Rightarrow \quad \dfrac{a}{s^2 - a^2}$

    
    

18. $\cosh(at) \quad \Rightarrow \quad \dfrac{s}{s^2 - a^2}$

    
    

19. $e^{at} \sin(bt) \quad \Rightarrow \quad \dfrac{b}{(s - a)^2 + b^2}$

    
    

20. $e^{at} \cos(bt) \quad \Rightarrow \quad \dfrac{s - a}{(s - a)^2 + b^2}$

    
    

21. $e^{at} \sinh(bt) \quad \Rightarrow \quad \dfrac{b}{(s - a)^2 - b^2}$

    
    

22. $e^{at} \cosh(bt) \quad \Rightarrow \quad \dfrac{s - a}{(s - a)^2 - b^2}$

    
    

23. $t^n e^{at}, \quad n = 1, 2, 3, \dots \quad \Rightarrow \quad \dfrac{n!}{(s - a)^{n+1}}$

    
    

24. $f(ct) \quad \Rightarrow \quad \dfrac{1}{c} F\left(\dfrac{s}{c}\right)$

    
    

25. $u_c(t) = u(t - c) \quad \Rightarrow \quad \dfrac{e^{-cs}}{s}$

    
    

26. $\delta(t - c) \quad \Rightarrow \quad e^{-cs}$

    
    

27. $u_c(t) f(t - c) \quad \Rightarrow \quad e^{-cs} F(s)$

    
    

28. $u_c(t) g(t) \quad \Rightarrow \quad e^{-cs} \mathcal{L}\{g(t + c)\}$

    
    

29. $e^{ct} f(t) \quad \Rightarrow \quad F(s - c)$

    
    

30. $t^n f(t), \quad n = 1, 2, 3, \dots \quad \Rightarrow \quad (-1)^n F^{(n)}(s)$

    
    

31. $\dfrac{1}{t} f(t) \quad \Rightarrow \quad \int_s^\infty F(u) \, du$

    
    

32. $\int_0^t f(v) \, dv \quad \Rightarrow \quad \dfrac{F(s)}{s}$

    
    

33. $\int_0^t f(t - \tau) g(\tau) \, d\tau \quad \Rightarrow \quad F(s) G(s)$

    
    

34. $f(t + T) = f(t) \quad \Rightarrow \quad \int_0^T e^{-st} f(t) \, dt \ \ \dfrac{1}{1 - e^{-sT}}$

    
    

35. $f'(t) \quad \Rightarrow \quad sF(s) - f(0)$

    
    

36. $f''(t) \quad \Rightarrow \quad s^2 F(s) - sf(0) - f'(0)$

    
    

37. $f^{(n)}(t) \quad \Rightarrow \quad s^n F(s) - s^{n-1} f(0) - s^{n-2} f'(0) \dots - sf^{(n-2)}(0) - f^{(n-1)}(0)$

    
    

3. Table Notes:

1. This list is not a complete listing of Laplace transforms and only contains some of the more commonly used Laplace transforms and formulas.
   Recall the definition of hyperbolic functions.

• $\cosh(t) = \dfrac{e^t + e^{-t}}{2}$
• $\sinh(t) = \dfrac{e^t - e^{-t}}{2}$

3. Be careful when using "normal" trig functions vs. hyperbolic functions. The only difference in the formulas is the "$+a^2$" for the "normal" trig functions becomes a "$-a^2$" for the hyperbolic functions!

4. Formula #4 uses the Gamma function, which is defined as

• $\Gamma(t) = \int_0^\infty e^{-x} x^{t-1} \, dx$

If $n$ is a positive integer, then:

• $\Gamma(n+1) = n!$

The Gamma function is an extension of the normal factorial function. Here are a couple of quick facts for the Gamma function

• $\Gamma(p+1) = p \Gamma(p)$
• $p(p+1)(p+2) \dots (p+n-1) = \left(\dfrac{\Gamma(p+n)}{\Gamma(p)}\right)$
• $\Gamma\left(\dfrac{1}{2}\right) = \sqrt{\pi}$

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