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Fourier Integrals, Heat Kernels and a One-Dimensional Heat Equation

I am having difficulty computing u(x,t), also interpretation when e -> 0
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COMMENT FROM STUDENT:
Thank you for your solution.
I don't queit understand how you get (2) from (1)?
what dose it mean e << x , does it mean that e is smaller than x?
how can you say that when e-> 0 (2) becomes pricise for all x?
is there any way actually get u(x,t) by integrating?

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Fourier Integrals, Heat Kernels and a One-Dimensional Heat Equation are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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