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Hello everybody

First I'd like to thank for the work all of you are developing with this forum. I found it for casuality but I'm sure since now it will be a perpetual partner.

I'm a Spanish Physics graduate and I am working about microwave guides and connectors for devices components in satellites and I have some trouble in my job. Although my questions's title it's not my trouble I think the solution could help for my investigation development.

My doubt is about how to change of variable in a Dirac Delta distribution. I know the usually called scaling property:

delta[f(x)]=Sum[delta(x-Xi)]/|f´(Xi)|

where Xi's are the roots of the function f(x). But my trouble is, for example, in the case that the function is as apparently innocent as

f(x)=x^2

because in this case the function has a double root Xi whose value is zero and this is a problem in the denominator of the expression, because

f´(Xi)=2*Xi

I'd like to receive ideas to solve this problem, although I have a possible way for the beginning.

If the function is f(x^2-a^2) with 'a' a real number, the solution is the well-known formula

delta[x^2-a^2]=1/(2·|a|)·[delta(x-a)+delta(x+a)]

How about if we take the limit 'a' tending to zero? I have no answer to this, but I think it could be an initial idea. I have looked at some books of calculus and I haven't found answer to this problem, but I recognize I have not read all the mathematical books that exist. I am sure my problem is that I have not read the development of this formula to know how adapt it to this case.

Thank you for to pay attention.

First I'd like to thank for the work all of you are developing with this forum. I found it for casuality but I'm sure since now it will be a perpetual partner.

I'm a Spanish Physics graduate and I am working about microwave guides and connectors for devices components in satellites and I have some trouble in my job. Although my questions's title it's not my trouble I think the solution could help for my investigation development.

My doubt is about how to change of variable in a Dirac Delta distribution. I know the usually called scaling property:

delta[f(x)]=Sum[delta(x-Xi)]/|f´(Xi)|

where Xi's are the roots of the function f(x). But my trouble is, for example, in the case that the function is as apparently innocent as

f(x)=x^2

because in this case the function has a double root Xi whose value is zero and this is a problem in the denominator of the expression, because

f´(Xi)=2*Xi

I'd like to receive ideas to solve this problem, although I have a possible way for the beginning.

If the function is f(x^2-a^2) with 'a' a real number, the solution is the well-known formula

delta[x^2-a^2]=1/(2·|a|)·[delta(x-a)+delta(x+a)]

How about if we take the limit 'a' tending to zero? I have no answer to this, but I think it could be an initial idea. I have looked at some books of calculus and I haven't found answer to this problem, but I recognize I have not read all the mathematical books that exist. I am sure my problem is that I have not read the development of this formula to know how adapt it to this case.

Thank you for to pay attention.

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