Is it possible to use complex contour integrals to solve gaussian integrals?

[aaaa202]

I have an integral of the form (from -inf to inf):

∫exp((a+ib)x²+icx)dx

How do I solve that?

I have tried setting y = √(a+ib)(x+ic/√(a+ib))

And you then get an integral of exp(-y²) times a constant. But I don't even know if this is legal since part of the exponentials are complex. Is it possible to obtain the result with my method?

 

[Fightfish]

Have you tried completing the square on the exponent?

 

[aaaa202]

what do you mean? Isn't that what I do by the introduction of my y?

 

[Fightfish]

Oh okay that didnt look clear to me as I expected you to leave it in the square form directly. I can't prove it off-hand, but the gaussian integration formula holds for complex a,b and c (where the exponent is in the form -ax^2 + bx + c) as long as the real part of a is positive

 

[Ray Vickson]

Oh okay that didnt look clear to me as I expected you to leave it in the square form directly. I can't prove it off-hand, but the gaussian integration formula holds for complex a,b and c (where the exponent is in the form -ax^2 + bx + c) as long as the real part of a is positive

What you want is not too hard to prove if you have a bit of complex function theory. If you take
I = \int_{-\infty}^{\infty} e^{-(a + ib)(x + iy)^2} \, dx, \: a > 0
you can write it in the form of a complex contour integral:
I = \int_{\Gamma} e^{- (a + ib) z^2} \, dz,
where $\Gamma$ is the line from -∞ + iy to +∞ + iy. This is the limit as M → ∞ of I_M, the integral from -M + iy to +M + iy. Since the integrand is an analytic function of z, get the same result I_M if we deform the contour into a path from -M+iy to -M, then from -M to +M, then from M to M+iy. The part from -M to +M goes along the real axis, and the parts from -M+iy to -M and from M to M+iy go to zero as M → ∞, so we are left with
I = \int_{-\infty}^{\infty} e^{-(a + ib) x^2} \, dx.
Now the usual method of squaring I and converting the double integral to polar coordinates applies essentially unchanged.