Dear MOs,

I need to calculate the inverse Laplace transform of the following function

$$ g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$

I have checked, among many others, the book "Table of Integral Transform, Vol. I". In P.267, Eq. (14) is for

$$ g(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}} $$

which is almost what I need. Other than this formula, I didn't find the one that I need. I have tried mathematica, which couldn't give an answer. I think the hope to find out the solution is quite small.

EIDT: here is some motivation of the problem.

Suppose the inverse transform gives us a function $f_a(t)$. I want to see the limit $$\lim_{a\rightarrow 0_+} f_a(t)=?$$

Can I simply do this:

$$ \lim_{a \rightarrow 0_+} \mathcal{L}^{-1}\left(g_a\right)(t) \stackrel{?}{=} \mathcal{L}^{-1}\left(\lim_{a\rightarrow 0_+} g_a\right)(t) = \mathcal{L}^{-1}\left(g_0\right)(t) =\frac{1}{\sqrt{\pi t}} + 2 e^{4t} \text{erfc}(-2\sqrt{t}) $$

Thank you very much for any hints!

Anand

Dear MOs,

I need to calculate the inverse Laplace transform of the following function

$$ g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$

I have checked, among many others, the book "Table of Integral Transform, Vol. I". In P.267, Eq. (14) is for

$$ g(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}} $$

which is almost what I need. Other than this formula, I didn't find the one that I need. I have tried mathematica, which couldn't give an answer. I think the hope to find out the solution is quite small.

EIDT: here is some motivation of the problem.

Suppose the inverse transform gives us a function $f_a(t)$. I want to see the limit $$\lim_{a\rightarrow 0_+} f_a(t)=?$$

Can I simply do this:

$$ \lim_{a \rightarrow 0_+} \mathcal{L}^{-1}\left(g_a\right)(t) \stackrel{?}{=} \mathcal{L}^{-1}\left(\lim_{a\rightarrow 0_+} g_a\right)(t) = \mathcal{L}^{-1}\left(g_0\right)(t) =\frac{1}{\sqrt{\pi t}} + 2 e^{4t} \text{erfc}(-2\sqrt{t}) $$

Are there some Lebesgue's dominated convergence theorems to use in complex analysis?

Thank you very much for any hints!

Anand

Dear MOs,

I need to calculate the inverse Laplace transform of the following function

$$ g(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$

I have checked, among many others, the book "Table of Integral Transform, Vol. I". In P.267, Eq. (14) is for

$$ g(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}} $$

which is almost what I need. Other than this formula, I didn't find the one that I need. I have tried mathematica, which couldn't give an answer. I think the hope to find out the solution is quite small.

EIDT: here is some motivation of the problem.

Suppose the inverse transform gives us a function $f_a(t)$. I want to see the limit $$\lim_{a\rightarrow 0_+} f_a(t)=?$$

Can I simply do this:

$$ \lim_{a \rightarrow 0_+} \mathcal{L}^{-1}\left(g_a\right)(t) \stackrel{?}{=} \mathcal{L}^{-1}\left(\lim_{a\rightarrow 0_+} g_a\right)(t) = \mathcal{L}^{-1}\left(g_0\right)(t) =\frac{1}{\sqrt{\pi t}} + 2 e^{4t} \text{erfc}(-2\sqrt{t}) $$

Thank you very much for any hints!

Anand

Dear MOs,

I need to calculate the inverse Laplace transform of the following function

$$ g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$

I have checked, among many others, the book "Table of Integral Transform, Vol. I". In P.267, Eq. (14) is for

$$ g(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}} $$

which is almost what I need. Other than this formula, I didn't find the one that I need. I have tried mathematica, which couldn't give an answer. I think the hope to find out the solution is quite small.

EIDT: here is some motivation of the problem.

Suppose the inverse transform gives us a function $f_a(t)$. I want to see the limit $$\lim_{a\rightarrow 0_+} f_a(t)=?$$

Can I simply do this:

$$ \lim_{a \rightarrow 0_+} \mathcal{L}^{-1}\left(g_a\right)(t) \stackrel{?}{=} \mathcal{L}^{-1}\left(\lim_{a\rightarrow 0_+} g_a\right)(t) = \mathcal{L}^{-1}\left(g_0\right)(t) =\frac{1}{\sqrt{\pi t}} + 2 e^{4t} \text{erfc}(-2\sqrt{t}) $$

Thank you very much for any hints!

Anand