How to solve an integral II

[Ad VanderVen]

TL;DR Summary

How to solve the following integral (in Maple notation): Int(y**k*exp(-u[0]*exp(-y)/a[0]-u[1]*exp(y)/a[1]),y=-infinity..infinity)?

How to solve the following integral (in Maple notation):

Int(y**k*exp(-u[0]*exp(-y)/a[0]-u[1]*exp(y)/a[1]),y=-infinity..infinity)

with 0<a[0], 0<u[0], 0<a[1], 0<u[1]?

 

[mathman]

Use LaTex!

 

[Ad VanderVen]

I am looking for a solution of the following integral

$$\int_{-\infty }^{\infty }\!{y}^{k}{{\rm e}^{-{\frac {u_{{0}} \, {{\rm e}^{-y}}}{a_{{0}}}}-{\frac {u_{{1}} \, {{\rm e}^{y}}}{a_{{1}}}}}}\,{\rm d}y$$
with $0<a_{0}$, $0<a_{1}$, $0<c_{0}$, $0<c_{1}$ and $k = 1, 2, 3, ...,$.

 

[BvU]

Now that we are offf the unanswered threads list anyway:

I find it hard to believe an analytical answer exists.

Doodled with $\ u_0=a_0 \ \ \& \ \ u_1 = a_1\$ so the integral becomes
$$ \int y^k\; e^{ -\cosh y} $$ and all the odd $k$ yield zero. What remains is interesting enough:

Wolframalpha does the work: for $k=2$ the bounds -4, 4 are already wide enough for 6 digits (0.615622) and of course 0, 5 gives the same result (but more digits !?):

with $k=4$:

(don't trust all digits, though:

and so on

But: with larger $k$ things become slippery:

So the natural questions are:

• Where does this come from and where is it going ?
• Can you make do with a numerical answer ?
• What precision do you need ?

 

[Charles Link]

Just so it doesn't become a stumbling point later, one minor correction: $\cosh{y}=\frac{e^y+e^{-y}}{2}$, with a 2 in the denominator.

 

[pasmith]

Now that we are offf the unanswered threads list anyway:

I find it hard to believe an analytical answer exists.

Doodled with $\ u_0=a_0 \ \ \& \ \ u_1 = a_1\$ so the integral becomes
$$ \int y^k\; e^{ -\cosh y} $$ and all the odd $k$ yield zero. What remains is interesting enough:

The general case is $$\int_{-\infty}^\infty (y-C)^k \mathrm{e}^{-\cosh y}\,dy$$ where $$\tanh C = \frac{a_0u_1 - a_1u_0}{a_0u_1 + a_1u_0}$$ It probably has to be done numerically.

 

[BvU]

Just so it doesn't become a stumbling point later, one minor correction: $\cosh{y}=\frac{e^y+e^{-y}}{2}$, with a 2 in the denominator.

So actually I doodled with $\ 2u_0=a_0 \ \ \& \ \ 2u_1 = a_1\$

The general case is

Can you show how you do that ?

 

[pasmith]

So actually I doodled with $\ 2u_0=a_0 \ \ \& \ \ 2u_1 = a_1\$
Can you show how you do that ?

That should be $$\int_{-\infty}^\infty (y-C)^k \mathrm{e}^{-D\cosh y}\,dy$$
with $C$ as before and $$D = 2 \sqrt{\frac{u_0u_1}{a_0a_1}}$$

 

[BvU]

Still don't see it ...

 

[pasmith]

Still don't see it ...

Use $e^{\pm y} = \cosh y \pm \sinh y$ and $$D \cosh (y + C) = D \cosh C \cosh y + D \sinh C \sinh y$$ to obtain $$Ae^y + Be^{-y} = (A + B) \cosh y + (A - B) \sinh y = D \cosh(y + C)$$
where $$D \cosh C = A + B),\qquad D\sinh C = A - B.$$ Then substitute $y + C = t$.

But one could go further and use $\int_{-\infty}^\infty f(y)\,dy = \int_0^\infty f(y) + f(-y)\,dy$ to obtain $$\int_0^\infty ((y - C)^k + (-1)^k (y + C)^k)\mathrm{e}^{-D\cosh y}\,dy$$ and expand the polynomials to obtain $$(y - C)^k + (-1)^k (y + C)^k = (-1)^k \sum_{r=0}^k \frac{k!}{(k-r)!r!}C^{k-r} (1 + (-1)^r)y^r$$ from which we see that the terms in odd $r$ vanish and we are left with a linear combination of $\int_0^\infty y^r \mathrm{e}^{-D\cosh y}\,dy$.

Now for large $D$ we can approximate this as $$\mathrm{e}^{-D} \int_0^\infty y^r \mathrm{e}^{-\frac12 Dy^2}\,dy$$ and the substitution $t = \frac12 D y^2$ will turn this into a Gamma function:
$$\int_0^\infty y^r \mathrm{e}^{-\frac12 D y^2}\,dy = \frac{1}{\sqrt{2D}} \left( \frac 2D\right)^{r/2} \int_0^\infty t^{\frac{r-1}2}\mathrm{e}^{-t}\,dt = \frac{1}{\sqrt{2D}} \left( \frac 2D\right)^{r/2}\Gamma\left( \frac{r+1}{2}\right).$$ Thus for large $D$ $$\int_{-\infty}^\infty (y - C)^k \mathrm{e}^{-D\cosh y}\,dy \approx \frac{(-1)^k \mathrm{e}^{-D}}{\sqrt{2D}} \sum_{r=0}^k \frac{k!}{(k-r)!r!} C^{k-r}(1 + (-1)^r) \left(\frac 2D\right)^{r/2} \Gamma\left( \frac{r+1}{2}\right).$$ To leading order only the $r=0$ term is relevant, yielding an estimate of $(-1)^kC^k \mathrm{e}^{-D}\sqrt{2\pi/D}$.