Integral transform

An integral transform maps a function to a new function by integration against a kernel.

The Mellin transform of $f:(0,\infty )\to \mathbb{C}$ iswhere the integral converges.

The Laplace transform is $\hat{f}(p)={\int }_0^{\infty }{e}^{-pt}f(t)dt$ and converts differentiation into multiplication by $p$ plus initial data.

Integration by parts giveswhenever the boundary term at infinity vanishes.

For $a\ge 0$, $\mathcal{L}\{f(t-a)H(t-a)\}(s)=e^{-as}\mathcal{L}\{f\}(s)$.

The delayed unit step has

The Heaviside function is zero before its switching time and one after it, allowing delayed signals to be written algebraically.

If $g$ has period $T$, then

The Laplace transform of a matrix-valued function is obtained by transforming each entry. A constant matrix may be taken outside on the corresponding side of the integral; in particular, $\mathcal{L}\{AB\}=A\mathcal{L}\{B\}$ when $A$ is constant.

Forthe derivative rule givesand hence $Y(s)=(sI-A{)}^{-1}(y_0+G(s))$ wherever the transforms converge and $sI-A$ is invertible.