Integrand with Lambert W -function

[Theia]

Hello!

I am modelling projectile motion under some non-simplified assumptions and I should obtain a some sort of 'solution' for the following horrible looking(?) integral

\(\displaystyle I(x, k) = \int_{-a}^x W(t) \sqrt[k]{t + a}\ \mathrm{d}t\),

where \(\displaystyle x \ge -a = -\mathrm{e}^{-1}\) and \(\displaystyle k \in \mathbb{Z}_+\). As for the \(\displaystyle x\), I can fix \(\displaystyle x = X\), that's not a problem for e.g. numerical integration.

But is there any other method to compute an approximation for \(\displaystyle I(x, k)\) than numerical methods? As far as I know, writing \(\displaystyle W(t)\) in terms of power serie is quite messy. But is there any other approachs?

Thank you! ^^

 

[I like Serena]

Hello!

I am modelling projectile motion under some non-simplified assumptions and I should obtain a some sort of 'solution' for the following horrible looking(?) integral

\(\displaystyle I(x, k) = \int_{-a}^x W(t) \sqrt[k]{t + a}\ \mathrm{d}t\),

where \(\displaystyle x \ge -a = -\mathrm{e}^{-1}\) and \(\displaystyle k \in \mathbb{Z}_+\). As for the \(\displaystyle x\), I can fix \(\displaystyle x = X\), that's not a problem for e.g. numerical integration.

But is there any other method to compute an approximation for \(\displaystyle I(x, k)\) than numerical methods? As far as I know, writing \(\displaystyle W(t)\) in terms of power serie is quite messy. But is there any other approachs?

Thank you! ^^

Hey Theia! ;)

Really! You expect something with the Lambert $W$ function to have a "nice" primitive?
It's a function that evades all regular analysis!
As I see it, it's just a placeholder for stuff that we can't deal with otherwise.
I think we can only compute approximations with numerical methods.
Sorry!

 

[Theia]

Hey Theia! ;)

Really! You expect something with the Lambert $W$ function to have a "nice" primitive?
It's a function that evades all regular analysis!
As I see it, it's just a placeholder for stuff that we can't deal with otherwise.
I think we can only compute approximations with numerical methods.
Sorry!

Indeed, you may be right. Thank you!

To be more clear, perhaps not 'a nice primitive', but rather 'a suitable starting point that could be used somehow'. :D Mainly, I'm asking, because in some cases, if one is patient enough to bash his head against the brick wall, there are some non-trivial tricks that may simplify the situation (e.g. in simple fluid planet model, where one needs to integrate something like power function times inverse error function). Perhaps I'll think over this a little bit first too... With a strong brick wall, of course! ;)

 

[Theia]

Okay, one day more brick walls.

Let's substitute \(\displaystyle t = q\mathrm{e}^q \quad \Rightarrow \quad \mathrm{d}t = \mathrm{e}^q(1+q)\mathrm{d}q\). So one obtains

\(\displaystyle I(x_q, k) = \int_{-1}^{x_q}q\sqrt[k]{q\mathrm{e}^q + \mathrm{e}^{-1}}\mathrm{e}^q(1+q)\mathrm{d}q.\)

Integrating by parts by choosing

\(\displaystyle \begin{align*} f &= q & g' &= \sqrt[k]{q\mathrm{e}^q + \mathrm{e}^{-1}}\mathrm{e}^q(1+q) \\
f' &= 1 & g &= \tfrac{k}{k+1} \sqrt[k]{\left( q\mathrm{e}^q + \mathrm{e}^{-1} \right)^{k+1}}
\end{align*}\)

one obtains

\(\displaystyle I(x_q, k) = \frac{kx_q}{k+1}\sqrt[k]{\left( x_q\mathrm{e}^{x_q} + \mathrm{e}^{-1} \right)^{k+1}} - \frac{k}{k+1} \int_{-1}^{x_q}\sqrt[k]{\left( q\mathrm{e}^{q} + \mathrm{e}^{-1} \right)^{k+1}}\mathrm{d}q.\)

No more special functions to disturb simple numerics! ^^