Analyzing an Integral Solution for Exponential Integral with Imaginary Numbers

[EngWiPy]

Hello,

Is this integral solvable in compact form?

$$\int_0^{\infty}e^{-j\nu\alpha}E_1^m(-j\nu)\,d\nu$$

where $E_1(.)$ is the exponential integral and $j=\sqrt{-1}$. I am trying to avoid solving it numerically because its solution won't be the end result.

Thanks

 

[George Jones]

Is this integral solvable in compact form?

I am unfamiliar with notation. What does the $m$ mean in $E^m_1$? The same as $E_1 \left( -i\nu \right)^m$, or something else? Once I understand the notation, I will ask Maple to have a go at it.

 

[EngWiPy]

I am unfamiliar with notation. What does the $m$ mean in $E^m_1$? The same as $E_1 \left( -i\nu \right)^m$, or something else? Once I understand the notation, I will ask Maple to have a go at it.

Yes, m is an integer and $E_1^m(x)=\left(E_1(x)\right)^m$

 

[George Jones]

Arbitrarily, I set $\alpha = 1$ and $m=3$, and Maple not did give a closed form solution for this special case.

 

[EngWiPy]

Hello,

I have this function in an integration $E_1^m(z)$, which is the exponential integral to the power $m$. I am looking to write it in a way, such that I can evaluate in compact form the following integration

$$\int_0^{\infty}e^{-jxt}E_1^m(-jt)\,dt$$

to avoid calculating it numerically, where $j=\sqrt{-1}$. Is there any way to do that?

Thanks in advance

 

[Charles Link]

It isn't clear to me what $E_1^m(z)$ is. Can you please write out this function in more detail.

 

[EngWiPy]

It isn't clear to me what $E_1^m(z)$ is. Can you please write out this function in more detail.

It is

$$\left(\int_z^{\infty}\frac{e^{-t}}{t}\,dt\right)^m$$