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MathematicalPhysicist
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I need to find the fourie transform of f(x)=N*exp(-ax^2/2).
(N and a are constants).
well ofcourse iv'e put into the next integral:
[tex]\int_{-\infty}^{\infty}f(x)exp(-ikx)dx[/tex]
Iv'e changed variables, that i will get instead of exp(-ax^2/2)exp(-ikx),
exp(-z^2)exp(-ikz*sqrt(2/a)), but that didn't helped very much, perhaps I need to use here the definition of dirac delta function to simplify this.
any hints?
there's also a question to simplify the next terms:
1.(x^2+3)d(x+5) where d is dirac's delta function.
2.d(2x-8)
3.d(x^2+x-2)
as for 2, what i did is: d(2x-8)=d(2(x-4))=d(x-4)/2
for 3 i think it equals:
d(x^2+x-2)=d((x-1)(x+2))=d(x-1)+d(x+2)
but i didnt use the integral definition, iv'e just used the fact that at x=1 or x=-2 the argument of the functional is zero and thus by definition it diverges there otherwise it equals zero, is this a sound reasoning for this question?
for 1, not sure, but if we use the integral definition then [tex]\int_{-\infty}^{\infty}f(x)(x^2+3)d(x+5)dx=f(-5)*28=28*\int_{-\infty}^{\infty}f(x)d(x+5)dx[/tex] then (x^2+3)d(x+5)=28*d(x+5), or something like this?
thanks in advance.
(N and a are constants).
well ofcourse iv'e put into the next integral:
[tex]\int_{-\infty}^{\infty}f(x)exp(-ikx)dx[/tex]
Iv'e changed variables, that i will get instead of exp(-ax^2/2)exp(-ikx),
exp(-z^2)exp(-ikz*sqrt(2/a)), but that didn't helped very much, perhaps I need to use here the definition of dirac delta function to simplify this.
any hints?
there's also a question to simplify the next terms:
1.(x^2+3)d(x+5) where d is dirac's delta function.
2.d(2x-8)
3.d(x^2+x-2)
as for 2, what i did is: d(2x-8)=d(2(x-4))=d(x-4)/2
for 3 i think it equals:
d(x^2+x-2)=d((x-1)(x+2))=d(x-1)+d(x+2)
but i didnt use the integral definition, iv'e just used the fact that at x=1 or x=-2 the argument of the functional is zero and thus by definition it diverges there otherwise it equals zero, is this a sound reasoning for this question?
for 1, not sure, but if we use the integral definition then [tex]\int_{-\infty}^{\infty}f(x)(x^2+3)d(x+5)dx=f(-5)*28=28*\int_{-\infty}^{\infty}f(x)d(x+5)dx[/tex] then (x^2+3)d(x+5)=28*d(x+5), or something like this?
thanks in advance.