- #1

FatPhysicsBoy

- 62

- 0

## Homework Statement

Having trouble understanding dirac deltas, I understand what they look like and how you can express one (i.e. from the limiting case of a gaussian) but for the life of me I can't figure out why the results of some integrals featuring dirac deltas equate to what they do.

## Homework Equations

N/A

## The Attempt at a Solution

An example of one particularly baffling to me is the following (given [itex]u=x+a[/itex], and [itex]du = dx[/itex]):

1) [itex]\int^{\infty}_{-\infty}\delta(x-u)f(u-a)du = f(x-a)[/itex],

especially when trying to compare to the 'standard' definition of how the dirac delta behaves under an integral:

2) [itex]\int^{\infty}_{-\infty}\delta(x-x_{0})f(x)dx = f(x_{0})[/itex],

in my mind it looks like 1) should be equal to [itex]f(u) = f(x+a)[/itex] because I look at it as [itex]\int^{\infty}_{-\infty}\delta(x-u)f(x)dx[/itex] and compare to 2).