Computing a Gaussian integral

[homer]

Homework Statement

Let $a,b$ be real with $a > 0$. Compute the integral
$$I = \int_{-\infty}^{\infty} e^{-ax^2 + ibx}\,dx.$$

Homework Equations

Equation (1):
$$\int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}$$

Equation (2):
$$-ax^2 + ibx = -a\Big(x - \frac{ib}{2a}\Big)^2 - \frac{b^2}{4a}$$

The Attempt at a Solution

Completing the square in $-ax^2 + ibx$ gives me Equation (2), so that my integral is now
$$I = e^{-b^2/4a}\int_{-\infty}^{\infty} e^{-a(x-ib/2a)^2}\,dx.$$
Making the substitution $u = \sqrt{a}(x-ib/2a)$ I get $du = \sqrt{a}\,dx$ so tht my integral becomes
$$I = \frac{1}{\sqrt{a}}\,e^{-b^2/4a}\int_{-\infty - ib/2\sqrt{a}}^{\infty - ib/2\sqrt{a}} e^{-u^2}\,du.$$

But this doesn't seem right.

 

[ZetaOfThree]

You've got it right. Just note that $-\infty - ib/2\sqrt{a}=-\infty$ and $\infty - ib/2\sqrt{a}=\infty$, and you're basically done.

 

[homer]

Thanks Zeta. Huge brain fart on my part in making it rigorous. The integral I was trying to compute is the limit of

$$I(R_1, R_2) = \int_{-R_1}^{R_2} e^{-ax^2 + bx}\,dx$$

as $R_1, R_2 \to \infty$. Then I can make a subsitution $z = \sqrt{a}(x-ib/2a)$ to get the integral

$$I(R_1,R_2) = \frac{1}{\sqrt{a}}\,e^{-b^2/4a}\int_{-\sqrt{a}R_1-ib/2\sqrt{a}}^{\sqrt{a}R_2 - ib/2\sqrt{a}}e^{-z^2}\,dz.$$

The integrand $e^{-z^2}$ is analytic on the entire complex plane, so the integral is path independent. So in particular I can take it on a contour consisting of:

(1) A straight line up from $z = -\sqrt{a}R_1 - ib/\sqrt{a}$ up to the real axis at point $z = -\sqrt{a}R_1$.

(2) A straight line on the real axis from $z = -\sqrt{a}R_1$ to $z = \sqrt{a}R_2$.

(3) A straight line down from the real axis at point $z = \sqrt{a}R_2$ to $z = \sqrt{a}R_2 - ib/2\sqrt{a}$.

Taking the limit as $R_1, R_2 \to \infty$ the integrals on contour sections (1) and (3) vanish since $\lvert z\rvert \to \infty$ and thus $e^{-z^2} \to 0$ on these two vertical lines. The integral on contour (2) then becomes

$$\lim_{R_1, R_2 \to \infty}\int_2 e^{-z^2}\,dz = \int_{-\infty}^{\infty} e^{-x^2}\,dx = \sqrt{\pi}.$$

Thus we have

\begin{align*}
\int_{-\infty}^{\infty} e^{-ax^2 + bx}\,dx
& = \lim_{R_1,R_2 \to \infty} I(R_1,R_2) \\
& =
\lim_{R_1,R_2 \to \infty} \frac{1}{\sqrt{a}}\,e^{-b^2/4a}\int_{-\sqrt{a}R_1-ib/2\sqrt{a}}^{\sqrt{a}R_2 - ib/2\sqrt{a}}e^{-z^2}\,dz \\
& = \frac{1}{\sqrt{a}}\,e^{-b^2/4a}\sqrt{\pi}.
\end{align*}

 

[homer]

Oops, contour (1) should be from $z = -\sqrt{a}R_1 - ib/2\sqrt{a}$ to $z = -\sqrt{a}R_1$.

 

[paranormal]

Since I'm integrating over the entire real line, I would have expected the limits of integration to be -\infty to \infty, not -\infty - ib/2\sqrt{a} to \infty - ib/2\sqrt{a}. Also, I'm not sure how to evaluate the integral of e^{-u^2} without using the value of \int_{-\infty}^{\infty} e^{-x^2}\,dx, which is not given in the problem.

As a scientist, it is important to be precise and thorough in your approach to problem-solving. In this case, the first step would be to clarify any uncertainties or ambiguities in the given problem. For example, are we assuming that b is a purely imaginary number? Is the value of \int_{-\infty}^{\infty} e^{-x^2}\,dx allowed to be used in our solution?

Once these details are clarified, we can proceed with solving the integral. One approach would be to use the method of contour integration, where we integrate the complex function e^{-az^2 + ibz} over a closed contour in the complex plane. This approach may involve using the Cauchy integral theorem or the residue theorem.

Another approach would be to use the properties of the Gaussian integral, such as the fact that it is equal to \sqrt{\pi} when the exponent is -x^2. We can try to manipulate the given integral into this form by completing the square and making appropriate substitutions. However, as noted in the attempt at a solution, this may lead to some uncertainties and difficulties in the evaluation of the integral.

Overall, as a scientist, it is important to carefully consider all possible approaches and solutions, and to clearly communicate any uncertainties or assumptions made in the process.