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- Thread starter asdf60
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So by the properties of the delta function, you must mean:

[tex]\int_a^b \delta(x) f(x) dx = \left{ \begin{array}{cc} f(0) & \mbox{if } a<0<b\\ -f(0) & \mbox{if } b<0<a\\ 0 & \mbox{otherwise} \end{array}[/tex]

The last line implies the function must be zero everywhere but x=0 (or, to be more specific, any continuous function must be zero for x≠0 to satisfy this property), and the other two imply it cannot be zero at x=0, so it must be discontinuous. In fact, you could even show there is no discontinuous function which satisfies the above conditions by showing that the value at x must actually be infinite.

[tex]\int_a^b \delta(x) f(x) dx = \left{ \begin{array}{cc} f(0) & \mbox{if } a<0<b\\ -f(0) & \mbox{if } b<0<a\\ 0 & \mbox{otherwise} \end{array}[/tex]

The last line implies the function must be zero everywhere but x=0 (or, to be more specific, any continuous function must be zero for x≠0 to satisfy this property), and the other two imply it cannot be zero at x=0, so it must be discontinuous. In fact, you could even show there is no discontinuous function which satisfies the above conditions by showing that the value at x must actually be infinite.

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Unfortunately, that is not the way the function is defined in this problem. The definition given is:

[tex]\int_a^b \delta(x) f(x) dx =f(0)[/tex]

where a = -1, and b = 1, always.

Heh, i can't figure out how to make the limits of the integration -1 and 1 in latex.

[tex]\int_a^b \delta(x) f(x) dx =f(0)[/tex]

where a = -1, and b = 1, always.

Heh, i can't figure out how to make the limits of the integration -1 and 1 in latex.

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Ok. Also, the function f must be continuous, right? You can define a series of gaussian functions f_{n}(x) that get narrower and narrower, but always have a value of 1 at x=0. All you need to show is that, for any continuous function d(x), there is some n above which the integral of d(x)f_{n}(x) is less than one.

And for bounds on an integral (also powers, subscripts, summation indices, etc) you need to put brackets around the bounds if they are more than one chatacter (click to see the code):

[tex]\int_{-1}^{\sum_{k=1}^{\infty} e^{-p_k}} \delta(x) = 1[/tex]

And for bounds on an integral (also powers, subscripts, summation indices, etc) you need to put brackets around the bounds if they are more than one chatacter (click to see the code):

[tex]\int_{-1}^{\sum_{k=1}^{\infty} e^{-p_k}} \delta(x) = 1[/tex]

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Thanks for the help, and sorry for wasting your time.

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