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.
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Laplace transform table
| S.No. | Function | Laplace transform |
| 1. | 1 | \[\frac{1}{s}\] |
| 2. | eat | \[\frac{1}{s-a}\] |
| 3. | e−at | \[\frac{1}{s+a}\] |
| 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) | \[\frac{1}{s^2}\] |
