Laplace Transform Table | Electricalvoice

Laplace Transform Table

Laplace transform is a mathematical tool that converts a function of a real variable to a function of a complex variable s (complex frequency). It is used for solving differential equations. Here is the Laplace transform table.

Laplace transform table

S.No. Function Laplace transform
1. 1

    \[\frac{1}{s}\]

2. eat

    \[\frac{1}{s-a}\]

3. eat

    \[\frac{1}{s+a}\]

4. tn , where n= 1,2,3,……

    \[\frac{n!}{s^{n+1}}\]

5. tp , p > -1

    \[\frac{\Gamma (p+1)}{s^{p+1}}\]

6. tn eat, where n= 1,2,3,……

    \[\frac{n!}{(s-a)^{n+1}}\]

7.

    \[\sqrt{t}\]

    \[\frac{\sqrt{\pi}}{{2s^{\frac{3}{2}}}}\]

8. tn-½ , where n= 1,2,3,……

    \[\frac{1 \times 3\times 5......\times (2n+1)\sqrt{\pi}}{{2^ns^{n+\frac{1}{2}}}}\]

9. sin(at)

    \[\frac{a}{s^2+a^2}\]

10. cos(at)

    \[\frac{s}{s^2+a^2}\]

11. t sin(at)

    \[\frac{2as}{(s^2+a^2)^2}\]

12. t cos(at)

    \[\frac{s^2-a^2}{(s^2+a^2)^2}\]

13. sin(at) − at cos(at)

    \[\frac{2a^3}{(s^2+a^2)^2}\]

14. sin(at) + at cos(at)

    \[\frac{2as^2}{(s^2+a^2)^2}\]

15. cos(at) − at sin(at)

    \[\frac{s(s^2-a^2)}{(s^2+a^2)^2}\]

16. cos(at) + at sin(at)

    \[\frac{s(s^2+3a^2)}{(s^2+a^2)^2}\]

17. sin(at + b)

    \[\frac{s(sin(b))+acos(b)}{s^2+a^2}\]

18. cos(at + b)

    \[\frac{s(cos(b))-asin(b)}{s^2+a^2}\]

19. sinh(at)

    \[\frac{a}{s^2-a^2}\]

20. cosh(at)

    \[\frac{s}{s^2-a^2}\]

21. eat sin(bt)

    \[\frac{b}{(s-a)^2+b^2}\]

22. eat cos(bt)

    \[\frac{s-a}{(s-a)^2+b^2}\]

23. eat sinh(bt)

    \[\frac{b}{(s-a)^2-b^2}\]

24. eat cosh(bt)

    \[\frac{s-a}{(s-a)^2-b^2}\]

25. t sinh(bt)

    \[\frac{2bs}{(s^2-b^2)^2}\]

26. t cosh(bt)

    \[\frac{s^2+b^2}{(s^2-b^2)^2}\]

Laplace Transform Properties Table

Let F(s) be the laplace transform of f(t) i.e.

    \[f(t)\xrightarrow[]{laplace\; transform}F(s)\]

S.No. Function Laplace transform
1. f(at)

    \[\frac{1}{a}F(\frac{s}{a})\]

2. f(t-a)

    \[e^{-as}F(s)\]

3. eat f(t)

    \[F(s-a)\]

4. tn f(t)

    \[(-1)^n\frac{\mathrm{d^2} F(s)}{\mathrm{d} t^2}\]

5.

    \[\frac{1}{t}f(t)\]

    \[\int_{s}^{\infty}F(u)du\]

6.

    \[\int_{0}^{t}f(v)dv\]

    \[\frac{F(s)}{s}\]

7.

    \[\int_{0}^{t}f(t-\tau)g(\tau)d\tau\]

    \[F(s)G(s)\]

8.

    \[\frac{\mathrm{d} f(t)}{\mathrm{d} t}=f_{t}^{'}\]

    \[sF(s)-f(0)\]

9.

    \[\frac{\mathrm{d^2} f(t)}{\mathrm{d} t^2}=f_{t}^{''}\]

    \[s^2F(s)-sf(0)-f^{'}(0)\]

10.

    \[\frac{\mathrm{d^n} f(t)}{\mathrm{d} t^n}=f_{t}^{(n)}\]

    \[s^nF(s)-s^{n-1}f(0)-s^{n-2}f^{'}(0).....-f^{(n-1)}(0)\]

General functions Laplace transform Table

S.No. Function Laplace transform
1. δ(t)

    \[1\]

2. δ(t-a)

    \[e^{-as}\]

3. u(t)

    \[\frac{1}{s}\]

4. u(t-a)

    \[\frac{e^{-as}}{s}\]

5. r(t)

    \[\frac{1}{s^2}\]

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