Solving the distribution function

[astro_enthu]

I want to solve following double integration. I am stuck from long time. Any intermediate solution will also be appreciated.

$\int$$\int$ dvb dvl (feven+fodd)

where,

feven = (r*(vl$^{2}$+vb$^{2}$))^(-2*$\beta$)*(potential - 0.5*(vl$^{2}$+vb$^{2}$+vr$^{2}$))^(-3*$\beta$+5.5), where $\beta$,potential and vr are some constant numbers

fodd = (1-$\eta$)*tanh(r*vl*cos(b)/1000)*feven

Where, $\eta$, r,b are constant numbers. Which means I want to marginalise (feven+fodd) over vl and vb.

Integration limits for vl and vb comes from energy equation. Limits are -[itex]\sqrt{2*potential - vb$^{2}$ - vr$^{2}$}[/itex]<=vl<=[itex]\sqrt{2*potential - vb$^{2}$ - vr$^{2}$}[/itex] and -[itex]\sqrt{2*potential - vr$^{2}$}[/itex]<=vb<=[itex]\sqrt{2*potential - vr$^{2}$}[/itex]

Also -3.5< =$\beta$ <1
and 0<= $\eta$ <=2
Any help will be highly appreciated

 

[jvicens]

.

Hello there,

Thank you for sharing your problem with us. It seems like you are trying to solve a complex double integration involving multiple variables and constants. Without knowing the exact values of the constants, it is difficult to provide a specific solution. However, I can give you some general advice on how to approach this problem.

Firstly, it is important to simplify the integrand as much as possible before attempting to integrate. In this case, you can simplify the expression (feven+fodd) by combining the terms and using properties of exponents. This will make the integration easier and more manageable.

Secondly, you can try using substitution to simplify the integrand further. Choose a variable to substitute and try to express the integrand in terms of that variable. This will help you reduce the number of variables in the integration and make it easier to solve.

Thirdly, it might be helpful to break down the double integration into two separate integrations, one for vl and one for vb. This will make it easier to apply the integration limits and solve each integration separately.

Lastly, do not hesitate to use online tools or software to help you with the integration. There are many resources available that can solve complex integrations and provide step-by-step solutions. You can also consult with your peers or a mentor for guidance and support.

I hope this information helps you in solving your problem. Keep persevering and don't give up, as solving complex integrations can be challenging but also rewarding. Good luck!