Complex Integral: Existence of Formula for 𝑒^(-𝑎𝑥2+𝑏𝑥)

[Yegor]

$$\int_0^\infty e^{-ax^2+bx} dx$$, a and b may be complex.
Does exist any formula for this integral?
Or for $$\int_{-\infty}^\infty e^{-ax^2+bx} dx$$?

 

[mathman]

ax²-bx=a(x-b/2a)²+(b/2)²/a

Your first integral then becomes an erf integral, which can only be done numerically. The second integral has a closed form solution - integral of Gaussian.

 

[lurflurf]

$$\int_0^\infty e^{-ax^2+bx} dx$$, a and b may be complex.
Does exist any formula for this integral?
Or for $$\int_{-\infty}^\infty e^{-ax^2+bx} dx$$?

$$\int_{-\infty}^\infty e^{-ax^2} dx={\sqrt{\frac{\pi}{a}}}}$$
Consider your integral multiply it by a constant of the form exp(c) where c let's you conplete the square of the quadratic. Then observe
$$\int_{-\infty}^\infty e^{-x^2} dx=\int_{-\infty}^\infty e^{-(x+y)^2} dx$$
for any constant y

 

[lurflurf]

ax²-bx=a(x-b/2a)²+(b/2)²/a

Your first integral then becomes an erf integral, which can only be done numerically. The second integral has a closed form solution - integral of Gaussian.

It is true that the first will involve erf, while the second will have nicer form. Yet erf is a closed form. Also closed form verses numerical solution is kind of silly any way. log(2) is a closed form, but if you want a number you have to "do it numerically". The issue has more do do with how many function one want to define tabulate and use. The distinction between an answer erf(1) and one of sin(1) is mostly historical.