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Topic: dirac delta function is not a TRUE function??
Replies: 11   Last Post: Jul 19, 2011 11:08 PM

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Posts: 409
Registered: 3/1/08
Re: dirac delta function is not a TRUE function??
Posted: Jul 18, 2011 4:53 PM
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On Jul 12, 3:48 am, MBALOVER <> wrote:
> Hi all,
> From WIKI, I learn that dirac delta function is not a true function.
> However, I do not understand the explanation in WIKI and still get
> confused, why it is not a TRUE function.
> Could you please help me to understand it?
> Thank you.

One way of looking at it is by analogy with vectors :

Let's take two 4 vectors :

[x1 , x2 , x3 ] = X
[y1 , y2 , y3 ]= Y

The dot product for such vectors , X*Y is (x1 * y1) + (x2 * y2) + (x3
* x3)
(geometrically it's the (length of of X * length of Y * (cosine of
the angle between X and Y)) )
These vectors can be considered as having 3 components , indexed for
example by :
X(1) = x1
X(2) = x2
X(3) = x3

Similarly , a function f :R -> R can be considered a vector having R
components , component p being f(p) .
The dot product for functions f and g is (integral for x from -oo to
+oo of f(x) * g(x) )
(the integral can be considered an infinitesimal summation of f(x) *
g(x) for each component x)

This dot product for functions introduces some problems , since
integrals , being infinite , need not converge ,and functions need not
be integrable ) . Also , when treating functions as vectors , we can
introduce some vectors that would not correspond to any normal
function .

Let's get back to our 3-vector example . Assume we want a 3-vector
X,such that for any Y
Y*X = Y(2) . Our answer is [0,1,0] . Now , in the space of functions ,
assume we want a "function"
G such that for any F , integral(F*G) = F(0) . Only one "function"
satisfies the requirement . By looking at our vector example ,
[0,1,0] , for Y(2) , we notice that G must be 0 everywhere , except
G(0) ,because integral (F*G) must always be F(0) . In 0 , G must have
such a value that the overall integral of G is 1 . But G(x) is 0 for
almost every x , G(0) cannot be an ordinary number .
G(0) is a "number" so large that it cause the integral of G to become
1 , even though G(x) is 0 for all non-zero X . G is the dirac-delta
"function" .

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