How Can I Perform Double Numerical Integration in MATLAB or Mathematica?

[EngWiPy]

I have the pdf of a random variable found from the characteristic function given by

$$f_X(\alpha)=\frac{1}{2\pi}\sum_{m=0}^Mj^m{K\choose m}\int_0^{\infty}e^{-jt(m+\alpha)}E_1^m(-jt)\,dt$$

where $j=\sqrt{-1}$ and $E_1(x)$ is the exponential integral. I need to find the CDF of the random variable $X$ which is given by

$$F_X(x)=\int_0^xf_X(\alpha)\,d\alpha$$

I can interchange the integrals, but I ended with two numerical integrations as well.

How can I do this in MATLAB? Is it easier to do in Mathematica?

 

[EngWiPy]

First, how can I evaluate $\int_0^{\infty}e^{-jt(m+\alpha)}E_1^m(-jt)\,dt$ numerically in MATLAB? Previously I used Mathematica because I felt it is easier to use for numerical integration, but now I have access to MATLAB only. Note that the result will be a function of $\alpha$ and not a number.

 

[kreil]

If you want the result in terms of alpha and not a number, then you want to do symbolic integration not numerical integration (the latter uses quadrature methods and produces a number).

Check out the int function in Symbolic Math Toolbox. You'll also need to use the expint function to form the integrand.

 

[EngWiPy]

If you want the result in terms of alpha and not a number, then you want to do symbolic integration not numerical integration (the latter uses quadrature methods and produces a number).

Check out the int function in Symbolic Math Toolbox. You'll also need to use the expint function to form the integrand.

OK, so, it's possible. I thought numerical integrations result only in numbers. I will check that out. Thanks