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I try to solve an integration as follows,

$$\int \frac{sy^{-1}}{(1+sy^{-1})} \text{exp}(-\sqrt{y})dy$$

as you can see its pretty complicated, and I get an answer like the following using Wolfram Alpha $$e^{-i \sqrt{s}} s \left(e^{2 i \sqrt{s}} \text{Ei}\left(-i \sqrt{s}-\sqrt{y}\right)+\text{Ei}\left(i \sqrt{s}-\sqrt{y}\right)\right)$$

My question is I dont understand where this imaginary part is coming from? Any thoughts?

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  • $\begingroup$ The result is twice the real part of a complex number. $\endgroup$
    – Ron Gordon
    Oct 30, 2014 at 15:36
  • $\begingroup$ You can try adding an "Assumptions-> s $\in$ Reals" in the Integrate[] function. $\endgroup$ Oct 30, 2014 at 15:38
  • $\begingroup$ @user_of_math is this the way it should be written Integrate[(sy^-1)/(1+sy^-1)* exp(-\sqrt(y)),{y}, Assumptions-> s \in Real] $\endgroup$
    – Tyrone
    Oct 30, 2014 at 15:40
  • $\begingroup$ i think $Ei$ stands for exponential integral $\endgroup$ Oct 30, 2014 at 15:42
  • $\begingroup$ yup thats true but I don't understand where the imaginary is coming from $\endgroup$
    – Tyrone
    Oct 30, 2014 at 15:43

1 Answer 1

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The complex form comes from the fact that, after subbing $y=u^2$, you get an integral of the form

$$\int du \frac{2 s u}{s+u^2} e^{-u} $$

This is a deceptively nasty integral. The only way to get something even remotely recognizable is to use partial fractions, which delivers complex numbers:

$$\frac{2 s u}{s+u^2} = s \left (\frac1{u-i \sqrt{s}} - \frac1{u+i \sqrt{s}} \right ) $$

The antiderivatives are then expressed in terms of those Ei functions.

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