How Do You Solve This Complex Quantum Mechanics Integral?

[IHateMayonnaise]

Homework Statement

This is part of a much larger problem, however currently I am stuck at the following integral:

$$<br /> \int_{-\infty}^{\infty}exp\left(k^2\left((\Delta x)^2-\frac{i \hbar t}{2m}\right ) + k(ix-2(\Delta x)^2 \bar{k}_x)\right)dk$$

Where obviously everything should be taken as a constant except the plane old k's.

Homework Equations

see (1) and (3)

The Attempt at a Solution

i tried to complete the square, followed by u/du substitution which yielded one of the messiest equations I've ever seen. There is an integral I found in a table that looks promising:

$$\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4a}+c\right)$$

However this does not include the imaginary unit therefore I do not believe it is much use to me.

Cookies for anyone who can get me started in the right direction.

Thanks yall!

IHateMayonnaise

 

[Avodyne]

Your formula with a, b, & c is valid for complex a, b, & c as long as the real part of a is positive.

I think the $k^2(\Delta x)^2$ term in the exponent has the wrong sign.

 

[IHateMayonnaise]

Your formula with a, b, & c is valid for complex a, b, & c as long as the real part of a is positive.

I think the $k^2(\Delta x)^2$ term in the exponent has the wrong sign.

Why is the formula valid only if the real part of a is positive? (Also, this is assuming that it is a definite integral from $-\infty$ to $\infty$?)

Good call on the $k^2(\Delta x)^2$ having the wrong sign! Thanks so much :)

Edit: Also I made a mistake in that integral from the table; I originally put:

$$\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4ac}+c\right)$$

but I meant:

$$\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4a}+c\right)$$