Help with heat equation dirac delta function?

[fredfuchs]

Homework Statement

The question was way too long so i took a snap shot of it
http://sphotos-h.ak.fbcdn.net/hphotos-ak-snc7/397320_358155177605479_1440801198_n.jpg

Homework Equations

The equations are all included in the snapshot

The Attempt at a Solution

So for question A I've done what the question said:

so g(x)^2 :

= 1/(\sqrt(2\pi)\delta)^2\iint\exp(-(x^2+y^2)/2\delta^2) dxdy
so i simplified to:
1/(\sqrt(2\pi)\delta)^2\iint\exp(-(r^2)/2\delta^2) rdrd\theta (0<\theta<2\pi) (0<r<inifity)
so i let z= r/2\delta^2

after all that i get to

1[\exp(\infty)-\exp(0)

its suppose to = 1, but i got -1 instead
i know it's wrong, but i don't know here, and the next 2 questions i don't have a clue how to start, can you help me?

 

[HallsofIvy]

Show us what you tried. Surely, the first part should be simple.

 

[fredfuchs]

I just added into what i did
sorry i did't know how to post mathematical equations until i looked it up before...
yeah, after that I pretty much have no clue...
Can you give me a few hints how to start the next few questions?