• Mathematica
• Hepth

#### Hepth

Gold Member
I have an integration to perform

$$\sqrt{\text{E1}^2-m^2} Log\left( \frac{m^2-m_B E1-m_B\sqrt{E1^2-m^2}}{m^2-m_B E1+m_B \sqrt{E1^2-m^2}}\right)$$
Code:
Sqrt[E1^2 - m^2]
log((m^2 - mB (Sqrt[E1^2 - m^2] + E1))/(
mB (Sqrt[E1^2 - m^2] - E1) + m^2))

over the region {E1,m,mB/2}

E1 is the variable. m is the minimum. mB/2 the max.

The integrand is valid over that region, and the two limits are both the points where the integrand crosses the axis.

I'd like an algebraic solution if possible. I can always go to numerical integration, but I'd prefer not to at this stage.

Mathematica won't solve it without limits, it just spits out the integral again. Same thing with the limits, but takes about 5 minutes to spit out the input.

Anyone have any non-numerical ideas?

the plot:
Code:
tmp2 = Sqrt[E1^2 - m^2]
Log[(m^2 - mB (Sqrt[E1^2 - m^2] + E1))/(
mB (Sqrt[E1^2 - m^2] - E1) + m^2)];
tmp3 = tmp2 /. {mB -> 5000., m -> 100., \[Lambda]2 -> 1.} //
FullSimplify;
Plot[tmp3, {E1, 100, 6000}, PlotRange -> {{0, 3000}, {0, 40000}}]

Last edited:
I think my limits are too high... nevermind.

Last edited:
Have you attempted to simplify the integrand by hand?

yeah, it simplifies a little, but doesn't change the problem. I think I made a mistake somewhere though, so give me another day before trying this. It'll probably change.

I simplified it some. Still a problem. Original equation updated.

The integration bounds are the same as the functions limits in the positive E1 region.

Usually, if Mathematica cannot solve the integral then that means that there is no analytical solution. I.e. Mathematica can do pretty much any integral that you would find in a book.

The only case where I have found a problem where an integral would not solve and I could do something about it was when there was some coordinate transformation that I could make which simplified it into something that was solvable.