Partial derivative + Integration

[ronaldoshaky]

Homework Statement

Part 1: I am trying to compute the partial derivative of exp (-ikx - ax^2) with respect to x

Part 2: i am trying to integrate, int (-ik - 2ax)*exp(-2ax^2) dx, with limits infinity and
- infinity

Homework Equations

part 1: see solution

part 2: i think integral of exp(-x^2)= sqrt(pi) should be used here.

The Attempt at a Solution

part 1: I worked out the partial derivative. I am not sure if its right.
(-ik - 2ax)*exp(-ikx - ax^2)

part 2: should i use integration by parts here. I don't know how to manipulate the expression.
can you multiply the power of an exponential [i.e.-2ax^2] in exp(-2ax^2) by 1/2a.

Thanks for your time.

 

[Nino MMP]

Part 1: The partial derivative of e^(-ikx-ax^2) with respect to x is (-ik-2ax)*e^(-ikx-ax^2).Part 2: Yes, you can use integration by parts here. Let u = -2ax^2 and dv = e^(-2ax^2)dx. Then du = -4ax dx and v = (1/2a)e^(-2ax^2). Therefore, the integral becomes (1/2a)*(1/2a)*e^(-2ax^2)|infinity to -infinity + 4a*int (1/2a)*e^(-2ax^2)dx. The second integral can be solved using the fact that integral of exp(-x^2)dx = sqrt(pi). Therefore, the final answer is (1/2a)*(1/2a)*e^(-2ax^2)|infinity to -infinity + sqrt(pi)*a.