Laplace Transform

A small list of the Transformation needed for an engineering student

22 flashcards · Mathematics · by Andrea

1. 111

   1s\frac{1}{s} \\s1​

2. xxx

   1s2\frac{1}{s^2}s21​

3. δ(x−a)\delta(x - a)δ(x−a)

   e−ase^{-as}e−as

4. eaxe^{ax}eax

   1s−a \frac{1}{s - a} \\s−a1​

5. sin⁡(ax)\sin(ax)sin(ax)

   as2+a2\frac{a}{s^2 + a^2} \\s2+a2a​

6. cos⁡(ax)\cos(ax)cos(ax)

   ss2+a2\frac{s}{s^2 + a^2} \\s2+a2s​

7. sinh⁡(ax)\sinh(ax)sinh(ax)

   as2−a2\frac{a}{s^2 - a^2}s2−a2a​

8. cosh⁡(ax)\cosh(ax)cosh(ax)

   ss2−a2\frac{s}{s^2 - a^2}s2−a2s​

9. xn,n∈Nx^n, n \in \mathbb{N}xn,n∈N

   n!sn+1\frac{n!}{s^{n+1}} \\sn+1n!​

10. f′(x)f'(x)f′(x)

    sF(s)−f(0)sF(s) - f(0)sF(s)−f(0)

11. ∫0xf(u) du\int_0^x f(u) \, du∫0x​f(u)du

    F(s)s\frac{F(s)}{s}sF(s)​

12. 1−e−axx\frac{1 - e^{-ax}}{x}x1−e−ax​

    ln⁡(s+as)\ln\left(\frac{s + a}{s}\right)ln(ss+a​)

13. sinh⁡(ax)ax\frac{\sinh(ax)}{ax}axsinh(ax)​

    1s2−a2\frac{1}{\sqrt{s^2 - a^2}}s2−a2 ​1​

14. 1−cos⁡(ax)ax\frac{1 - \cos(ax)}{ax}ax1−cos(ax)​

    ss2+a2\frac{s}{s^2 + a^2}s2+a2s​

15. e−ax2e^{-ax^2}e−ax2

    πa⋅es24a⋅erfc(s2a)\sqrt{\frac{\pi}{a}} \cdot e^{\frac{s^2}{4a}} \cdot \text{erfc}\left(\frac{s}{2\sqrt{a}}\right)aπ​ ​⋅e4as2​⋅erfc(2a ​s​)

16. 1x\frac{1}{\sqrt{x}}x ​1​

    πs\sqrt{\frac{\pi}{s}}sπ​ ​

17. u(x−a)u(x - a)u(x−a)

    e−ass\frac{e^{-as}}{s}se−as​

18. e−axe^{-ax}e−ax

    1s+a\frac{1}{s+a}s+a1​

19. xe−atxe^{-at}xe−at

    1(s+a)2\frac{1}{(s+a)^2}(s+a)21​

20. e−atsin⁡(ωx)e^{-at} \sin(\omega x)e−atsin(ωx)

    w(s+a)2+ω2 \frac{w}{(s+a)^2 + \omega^2} (s+a)2+ω2w​

21. e−atcos⁡(ωt)e^{-at} \cos(\omega t)e−atcos(ωt)

    s+a(s+a)2+ω2\frac{s+a}{(s+a)^2+\omega^2}(s+a)2+ω2s+a​