Laplace And Inverse Laplace Transforms

Let a function $f$ be defined for $t\ge 0$. Then, Laplace Transform of $f$ is defined as:

for those values of $s$ where this integral exists.

Also, the inverse Laplace Transform is defined as:

First Shifting Theorem: If $\mathcal{L}(f(t))=F(s)$, then $\mathcal{L}(e^{at}f(t))=F(s-a)$.

Laplace Transforms Of Elementary Functions

The unit step function $u_a(t)$ is defined as:

Laplace Transform of $u_a(t)$,

Second Shifting Theorem: If $\mathcal{L}(f(t))=F(s)$, then

Theorem: Let $f(t)$ be continuous for $t\ge 0$ and is of exponential order. Further, suppose that $f$ is differentiable with $f^{\prime }$ piecewise continuous in $[0,\mathrm{\infty })$. Then,

Also,

$\mathcal{L}\left(f^{(n)}\right)$=$s^n\mathcal{L}(f)$-$s^{n-1}f(0)$-$s^{n-2}{f}^{(1)}(0)$-$\cdots$-$f^{(n-1)}(0)$

Theorem: Let $F(s)$ be the Laplace Transform of $f$. If $f$ is piecewise continuous in $[0,\mathrm{\infty })$ and is of exponential order, then

Theorem: Let $F(s)$ be the Laplace Transform of $f$, then

Theorem: Let $F(s)$ be the Laplace Transform of $f$ and the limit of $f(t)/t$ exists as $t\to 0^{+}$, then

Initial Value Problems

Theorem for Periodic Functions: Let $f:[0,\mathrm{\infty })\to \mathbb{R}$ be a periodic function with period $T>0$, i.e., $f(t+T)=f(t)\mathrm{\forall }t\ge 0$. Then,

Convolution: Let $f$ and $g$ be two functions defined in $[0,\mathrm{\infty })$.

Then, the convolution of $f$ and $g$,is defined by:

Convolution Theorem: The convolution $f\ast g$ has the Laplce Transform property:

Laplace And Inverse Laplace Transforms

Initial Value Problems