0
$\begingroup$

I have no idea how to evaluate this integral. Tried Wolfram but it gives a erfc function which I have no knowledge about. It is actually a part of a bigger integral which I was trying to evaluate algebraically. Could anyone help me out?

$\int_{0}^\infty e ^{ix^2-x} $

  • 2
    Complete the square and use error function? – 2017-01-20
  • 0
    You do not solve an integral, but you evaluate or compute it :D – 2017-01-20
  • 0
    Thank you for correcting :) @SimpleArt I don't really understand what you mean with completing the square, also not familiar with the error function. I was hoping to evaluate the integral algebraically. – 2017-01-20
  • 0
    Complete the square in the argument. Use a good u-substitution. Do a lot of algebra. Cleverly apply the error function. This integral is a lot of work, with a less than stellar result. Unless you need an expression for some reason, computational methods are more worth your time. – 2017-01-20
  • 0
    Thank you SimplyBeautifulArt and Kayjex. Kind of feel embarrassed that I didn't know about the 'completing the square' trick. This will help met to get a step further. I actually need to prove that when I evaluate some integrals of that form that the sum is equal to an integer. – 2017-01-20

1 Answers 1

1

Maple's answer is $$ \left( -1/4-i/4 \right)\sqrt {2\pi}\;{{{\rm e}^{i/4}}} \left( {\rm erf} \left( \left( 1/4+i/4 \right) \sqrt {2}\right)-1 \right) $$ where $\rm erf$ is the error function. I don't think there's an answer more elementary than that.

  • 0
    I need to sum 8 different integrals of the same form but slightly different. The sum of the integrals needs to be an integer. They all have an error function when I evaluate them. Is it possible to get an integer when summing error functions? – 2017-01-20
  • 0
    This result can also be written n terms of the [Fresnel integrals](http://mathworld.wolfram.com/FresnelIntegrals.html) (see eq 1). This has the advantage of not requiring the $e^{i\pi /4}$ phase factor; one pays a price in that both Fresnel integrals are required, but this seems a small complaint. – 2017-01-20
  • 0
    Sure, for example $\rm erf(x)-\rm erf(x)=0$. Some more useful identities are $\rm erf(-x)=-\rm erf(x)$, $\rm erf(z*)=\rm erf(z)*$ or $\rm erf(z)=\rm erf(iz)/i$. I would recommend to aks an additional question which contains this sum of the 8 integrals. – 2017-01-20