Functions of Laplace transform

Functions of Laplace transform

$$1. \ L\left[ 1 \right] = \frac 1s$$
$$2. \ L\left[ e^{at} \right] = \frac {1}{s-a}$$
$$3. \ L\left[ t^n \right] = \frac {n!}{s^{n+1}} , n=1,2,3,\cdots$$
$$4. \ L\left[ t^p \right] = \frac {\Gamma(p+1)}{s^{p+1}} , p>-1$$
$$5. \ L\left[ \sqrt {t} \right] = \frac {\sqrt {\pi}}{2s^{\frac 32}}$$
$$6. \ L\left[ t^{n- \frac 12} \right] = \frac {1\cdot 3\cdot 5\cdots (2n-1)\sqrt {\pi}}{2^n s^{n+\frac 12}} , n=1,2,3,\cdots$$
$$7. \ L\left[ \sin at \right] = \frac {a}{s^2 + a^2}$$
$$8. \ L\left[ \cos at \right] = \frac {s}{s^2 + a^2}$$
$$9. \ L\left[ t\sin at \right] = \frac {2as}{(s^2 + a^2)^2}$$
$$10. \ L\left[ t\cos at \right] = \frac {s^2-a^2}{(s^2 + a^2)^2}$$
$$11. \ L\left[ \sin at -at \cos at \right] = \frac {2a^3}{(s^2 + a^2)^2}$$
$$12. \ L\left[ \sin at +at \cos at \right] = \frac {2as^2}{(s^2 + a^2)^2}$$
$$13. \ L\left[ \cos at -at \sin at \right] = \frac {s(s^2-a^2)}{(s^2 + a^2)^2}$$
$$14. \ L\left[ \cos at +at \sin at \right] = \frac {s(s^2+3a^2)}{(s^2 + a^2)^2}$$
$$15. \ L\left[ \sin (at+b) \right] = \frac {s \sin b + a \cos b}{s^2 + a^2}$$
$$16. \ L\left[ \cos (at+b) \right] = \frac {s \cos b – a \sin b}{s^2 + a^2}$$
$$17. \ L\left[ \sinh at \right] = \frac {a}{s^2 – a^2}$$
$$18. \ L\left[ \cosh at \right] = \frac {s}{s^2 – a^2}$$
$$19. \ L\left[ e^{at} \sin bt \right] = \frac {b}{(s-a)^2+b^2}$$
$$20. \ L\left[ e^{at} \cos bt \right] = \frac {s-a}{(s-a)^2+b^2}$$
$$21. \ L\left[ e^{at} \sinh bt \right] = \frac {b}{(s-a)^2-b^2}$$
$$22. \ L\left[ e^{at} \cosh bt \right] = \frac {s-a}{(s-a)^2-b^2}$$
$$23. \ L\left[ t^ne^{at} \right] = \frac {n!}{(s-a)^{n+1}}, n=1,2,3,\cdots$$
$$24. \ L\left[ f(ct) \right] = \frac 1c F \left( \frac sc \right)$$
$$25. \ L\left[ u_c(t)=u(t-c) \right] = \frac {e^{-cs}}{s}$$
$$26. \ L\left[ \delta (t-c) \right] = e^{-cs}$$
$$27. \ L\left[ u_c(t)f(t−c) \right] = e^{-cs} F(s)$$
$$28. \ L\left[ u_c(t)g(t) \right] = e^{-cs} L [ g(t+c) ]$$
$$29. \ L\left[ e^{ct}f(t) \right] = F(s-c)$$
$$30. \ L\left[ t^n f(t) \right] = (-1)^nF^{(n)}(s), n=1,2,3,\cdots$$
$$31. \ L\left[ \frac 1t f(t) \right] = \int\limits_s^{\infty}F(u)du$$
$$32. \ L\left[ \int\limits_0^t f(v)dv \right] = \frac {F(s)}{s}$$
$$33. \ L\left[ \int\limits_0^t f(t-T)g(T) dT \right] = F(s)G(s)$$
$$34. \ L\left[ f(t+T)=f(t) \right] = \frac {\int\limits_0^T e^{-st} f(t)dt}{1-e^{-sT}}$$
$$35. \ L\left[ f'(t) \right] = sF(s)-f(0)$$
$$36. \ L\left[ f”(t) \right] = s^2F(s)-sf(0)-f'(0)$$
$$37. \ L\left[ f^{(n)}(t) \right] = s^nF(s)-s^{n-1}f(0)-s^{n-2}f'(0) \cdots sf^{(n-2)}(0)-f^{(n-1)}(0)$$

Solution:

$$F(s)= 6 \frac {1}{s-(-5)} + \frac {1}{s-3} +5 \frac {3!}{s^{3+1}}-9 \frac 1s$$
$$F(s)= \frac {6}{s+5} + \frac {1}{s-3} + \frac {30}{s^4} – \frac 9s$$

Solution:

$$H(s)= 3 \frac {2}{s^2 – 2^2} + 3 \frac {2}{s^2 + 2^2}$$
$$H(s)= \frac {6}{s^2-4} + \frac {6}{s^2+4}$$