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I am trying to understand how I can numerically evaluate the imaginary error function with the incomplete gamma function for all $x \in \mathbb{R}$. I found here that the imaginary error function can be expressed as:

$$erfi(x) = \frac{\sqrt{-x^2}}{x}\big( Q( \frac{1}{2}, -x^2, \infty) -1 \big) $$

Where $Q$ is the regularized incomplete gamma function. For $x \in \mathbb{R}$ this equates to:

$$erfi(x) = i \big( Q( \frac{1}{2}, -x^2, \infty) -1 \big) $$

Now, I can evaluate the incomplete Gamma function, but using this method I won't get a real data type back. This confuses me, since for instance, another representation of the imaginary error function is:

$$erfi(x) = \frac{2}{\sqrt{\pi}} e^{x^{2}} D(x) $$

Where $D$ is the Dawson Function. This is entirely real valued! Also the scipy implementation gives me a real value too. [Yes I see the reference code, but it is pretty much impossible to follow. Besides, I want to understand how to do it myself.]

How can I implement the erfi function using the incomplete Gamma representation and still get a real value back?

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There are two points to note:

  1. The incomplete $\Gamma(1/2, -x^2)$ function is complex.

  2. $\frac{\sqrt{-x^2}}{x}$ is $-i$ except for real $x>0$.

Here is an example for $x=-2$

$$\mathrm{erfi(-2)} = -18.56480241$$ $$\Gamma(1/2,-4) = 1.772453851-32.90525553i$$ $$Q(1/2,-4) = \Gamma(1/2,-4)/\sqrt{\pi}=1.000000000-18.56480241 i$$

$$\Big(Q(1/2,-4)-1\Big)\frac{\sqrt{-4}}{-2} = (-18.56480241 i) \times (-i) = -18.56480241$$

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  • $\begingroup$ It is complex? Do you have a reference? I figured I could split the Gamma integral into positive and negative parts. Numerical Recipes then has some routines for evaluating the parts, but I guess I was wrong. $\endgroup$ – The Dude Aug 11 '16 at 13:50
  • $\begingroup$ Just enter Gamma(1/2, -4) into Wolfram Alpha! $\endgroup$ – gammatester Aug 11 '16 at 13:54
  • $\begingroup$ Yes, I did that. And you are right. But how do they do it? $\endgroup$ – The Dude Aug 11 '16 at 13:59
  • $\begingroup$ Althougj not exactly justified: A hint to the complex values comes from the integral representation $$\Gamma(1/2, -4) = \int_{-4}^\infty \frac{ e^{-t}}{\sqrt{t}} dt$$ where you take square roots from negative numbers. $\endgroup$ – gammatester Aug 11 '16 at 14:07
  • $\begingroup$ Okay. I figured I could do: $$\Gamma(\frac{1}{2}, -4) = \int_{0}^{\infty} \frac{e^{-t}}{\sqrt{t}} dt -\int_{0}^{4} \frac{e^{-t}}{\sqrt{t}} dt $$ But now you speak of negative square roots...hmmm... $\endgroup$ – The Dude Aug 11 '16 at 14:15

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