Laplace transform

In mathematics and in particular, in functional analysis, the Laplace transform of a function f(t) defined for all real numbers t ≥ 0 is the function F(s), defined by:

mathcal{L}{ f } mathcal{L}{ g }

Common transforms

• $n$th power

: mathcal{L}{,t^n} = rac {n!}{s^{n+1}}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Exponential

: mathcal{L}{,e^{-at}} = rac {1}{s+a}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Sine

: mathcal{L}{,sin(omega t)} = rac {omega}{s^2 + omega^2}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Cosine

: mathcal{L}{,cos(omega t)} = rac {s}{s^2 + omega^2}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Hyperbolic sine

: mathcal{L}{,sinh(bt)} = rac {b}{s^2-b^2}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Hyperbolic cosine

: mathcal{L}{,cosh(bt)} = rac {s}{s^2 - b^2}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Natural logarithm

: mathcal{L}{,ln(t)} = - rac{ln(s)+gamma}{s}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• nth root

: mathcal{L}{,sqrt{t}} = s^{-rac{n+1}{n}} cdot Gammaleft(1+rac{1}{n} ight)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Bessel function of the first kind

: mathcal{L}{,J_n(t)} = rac{left(s+sqrt{1+s^2} ight)^{-n}}{sqrt{1+s^2}}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Modified Bessel function of the first kind

: mathcal{L}{,I_n(t)} = rac{left(s+sqrt{s^2-1} ight)^{-n}}{sqrt{s^2-1}}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Error function

: mathcal{L}{,operatorname{erf}(t)} = {e^{s^2/4} operatorname{erfc} left(s/2 ight) over s}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• Periodic Function period $T$

: mathcal{L}{ f }

~ ~ ~ ~ ~ ~ ~ ~ ~ ~