Thread: Need a Laplace inversion formula

1. Need a Laplace inversion formula

Is there any Laplace inversion formula for any of these equations

$f(s)=\frac{1}{a\sqrt{s+b}+c\sqrt{s/d+b}}$
or
$
f(s)=\frac{\exp{(-k\sqrt{s+b})}}{a\sqrt{s+b}+c\sqrt{s/d+b}}
$

the inversion is easy if $d=1$, but I want to find the general form. Thank you.

2. Originally Posted by Ehsan
Is there any Laplace inversion formula for any of these equations

$f(s)=\frac{1}{a\sqrt{s+b}+c\sqrt{s/d+b}}$
or
$
f(s)=\frac{\exp{(-k\sqrt{s+b})}}{a\sqrt{s+b}+c\sqrt{s/d+b}}
$

the inversion is easy if $d=1$, but I want to find the general form. Thank you.
Well if you take the factor of 1/d out of the square root then you get a form that is equivalent to if d = 1. So if that form is easy for you, there you go.

3. I cannot invert it taking the factor of 1/d out

if d=1 then the square roots become identical (which makes it easy to invert) but if I take the factor 1/d out of the square root the equations become:

$
f(s)=\frac{1}{a\sqrt{s+b}+\frac{c}{\sqrt{d}}\sqrt{ s+bd}}
$

or
$
f(s)=\frac{\exp{(-k\sqrt{s+b})}}{a\sqrt{s+b}+\frac{c}{\sqrt{d}}\sqrt {s+bd}}
$

which I cannot invert.

4. Originally Posted by Ehsan
if d=1 then the square roots become identical (which makes it easy to invert) but if I take the factor 1/d out of the square root the equations become:

$
f(s)=\frac{1}{a\sqrt{s+b}+\frac{c}{\sqrt{d}}\sqrt{ s+bd}}
$

or
$
f(s)=\frac{\exp{(-k\sqrt{s+b})}}{a\sqrt{s+b}+\frac{c}{\sqrt{d}}\sqrt {s+bd}}
$

which I cannot invert.
Whoops! Right you are. I'll have a think while I'm asleep (which is very soon).