Help with heat equation dirac delta function?

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SUMMARY

The discussion focuses on solving a heat equation involving the Dirac delta function, specifically addressing the mathematical simplification of the integral expression. The user attempted to evaluate the integral using polar coordinates, resulting in an incorrect outcome of -1 instead of the expected value of 1. The conversation highlights the need for clarification on the proper approach to the subsequent questions related to the heat equation.

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  • Understanding of the heat equation and its applications in physics.
  • Familiarity with the Dirac delta function and its properties.
  • Knowledge of polar coordinates and their use in double integrals.
  • Basic skills in mathematical analysis and integral calculus.
NEXT STEPS
  • Study the properties of the Dirac delta function in the context of differential equations.
  • Learn about the method of separation of variables for solving the heat equation.
  • Explore the use of polar coordinates in evaluating double integrals.
  • Review examples of heat equation solutions involving initial conditions and boundary value problems.
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Students studying partial differential equations, mathematicians focusing on mathematical physics, and anyone seeking to understand the application of the Dirac delta function in heat equations.

fredfuchs
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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 :

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

after all that i get to

[tex]1[\exp(\infty)-\exp(0)[/tex]

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

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