Find power series if you know its laplace transformation

[skrat]

Homework Statement

a) Determine power series $\sum _{n=0}^{\infty }a_nt^n$ if you know that its laplace transformation is $-s^{-1}e^{-s^{-1}}$
b) Determine function $g$ that this power series will be equal to $J_0(g(t))$

Homework Equations

The Attempt at a Solution

Hmmm, I am having some troubles with this laplace transformation in part a).

Well, I know that Laplace transformation of Heaviside function $H_C(t)$ is $\frac{1}{s}e^{-Cs}$

Knowing this I get almost the same as the problem says: $-H_1$ ---> $-\frac{1}{s}e^{-s}$. But I have absolutely NO idea what to do to get $\frac{1}{s}$ in the exponent function. If I just power both sides of the equation everything else collapses...

So how can I deal with this?

 

[vela]

Expand the exponential as a series then invert the transform term by term.

 

[skrat]

Hmmmm...

$-\frac{1}{s}e^{-1/s}=-\frac{1}{s}(1-\frac{1}{s}+\frac{1}{2}(\frac{1}{s})^2-\frac{1}{3}(\frac{1}{s})^3+\frac{1}{4}(\frac{1}{s})^4- ...)=-\frac{1}{s}+(\frac{1}{s})^2-\frac{1}{2}(\frac{1}{s})^3+\frac{1}{3}(\frac{1}{s})^4-\frac{1}{4}(\frac{1}{s})^5+ ...$

Using inverse Laplace transformation gives me:

$-1+t-\frac{1}{4}t^2+\frac{1}{18}t^3-\frac{1}{96}t^4+\frac{1}{600}t^5-\frac{1}{4320}t^6+...$

BUT I can't fine a way to include that -1 into series:

$-1+\sum _{k=1}^{\infty }\frac{(-1)^{k+1}}{(k+1)!-k!}t^k$

 

[vela]

You didn't expand the exponential function correctly. It should be n! in the denominator, not n.

 

[skrat]

uh, jup, you are right.

The result is $\sum _{k=0}^{\infty }\frac{(-1)^{k+1}}{(k!)^2}t^k$.

Thank you, vela!