How to Solve the Scalar Density Integral in Spherical Coordinates?

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Discussion Overview

The discussion revolves around solving a scalar density integral in spherical coordinates, specifically the integral involving the dispersion relation and temperature. Participants explore the mathematical formulation and potential approaches to the problem.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant presents the integral to be solved, involving the dispersion relation and temperature.
  • Another participant questions how to add a scalar to a vector, suggesting that the derivative of the exponential term might be useful.
  • A later reply corrects the expression for the dispersion relation, indicating a misunderstanding in the initial post.
  • Another participant suggests trying spherical coordinates as a potential method for solving the integral.

Areas of Agreement / Disagreement

Participants do not appear to reach consensus on the method to solve the integral, and multiple approaches are suggested without resolution.

Korbid
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hi!

i need to solve this integral:

[itex]\rho_s=\int (m/\omega)e^{-\omega/T}d{\vec k}[/itex]
where [itex]\omega=\sqrt{m^2+{\vec k}}[/itex] is the dispersion relation, T is the temperature of the system and m the mass of a particle

Thank you!
 
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How do you add a scalar to a vector?

If k would be a scalar, the derivative of ##e^{-\omega / T}## looks promising.
 
Sorry! I was wrong!

[itex]\omega = \sqrt{m^2 + \vec{k}^2}[/itex]

However, i still can't solve it.
 
Did you try spherical coordinates?
 

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