Transformada de Laplace

• $F(s)=$$\mathscr{L}\{f(t)\}=$


  $\lim_{b \to \infty} \int_{0}^{b}e^{-st}f(t)dt$
• $\mathscr{L}\{1\}=$


  $\frac{1}{s} \space\space (s>0)$
• $\mathscr{L}\{e^{at}\}=$


  $\frac{1}{s-a} \space\space(s>a)$
• $\mathscr{L}\{sin(at)\}=$


  $\frac {a}{s^2+a^2} \space\space(s>0)$
• $\mathscr{L}\{cos(at)\}=$


  $\frac{s}{s^2+a^2} \space\space (s>0)$
• $\mathscr{L}\{t\}=$


  $\frac{1}{s^2}\space\space(s>0)$
• $\mathscr{L}\{t^{n-1}\}=$


  $\frac{(n-1)!}{s^{n}}\space\space(s>0)$
• $\mathscr{L}\{\sqrt{t}\}=$


  $\frac{1}{2} \sqrt{\pi} s^\frac{3}{2}\space\space(s>0)$
• $\mathscr{L}\{\frac{1}{\sqrt{t}}\}=$


  $\sqrt{\pi} s^\frac{1}{2}\space\space(s>0)$
• $\mathscr{L}\{t^{n-\frac{1}2{}}\}=$


  $\frac{(1)(3)(5)...(2n-1)\sqrt{\pi}}{2^{n}}\cdot s^{-n-\frac{1}{2}}\space\space(s>0)$
• $\mathscr{L}\{senh(at)\}=$


  $\frac{a}{s^{2}-a^{2}}\space\space\space(s>|a|)$
• $\mathscr{L}\{cosh(at)\}=$


  $\frac{s}{s^2-a^2}\space\space\space(s>|a|)$
• $\mathscr{L}\{tsen(at)\}=$


  $\frac {2as}{(s^2+a^2)^2}\space\space\space(s>0)$
• $\mathscr{L}\{tcos(at)\}=$


  $\frac{s^2-a^2}{(s^2+a^2)^2}\space\space\space(s>0)$
• $\mathscr{L}\{t^{n-1}e^{at}\}=$$\space\space\space;(n=1,2...)$


  $\frac{(n-1)!}{(s-a)^n}\space\space\space(s>a)$
• $\mathscr{L}\{sen(at)-atcos(at)\}=$


  $\frac{2a^3}{(s^2+a^2)^2}\space\space(s>0)$
• $\mathscr{L}\{f(t)\}=$


  $F(s)$
• $\mathscr{L}\{f'(t)\}=$


  $sF(s)-f(0)$
• $\mathscr{L}\{f''(t)\}=$


  $s^{2}F(s)-sf(0)-f'(0)$
• $\mathscr{L}\{f'''(t)\}=$


  $s^{3}F(0)-s^{2}f(0)-sf'(0)-f''(0)$