Need reference about the derivatives of Direc Delta function...

Specificly, I need to find the function values of the nth derivative of
Direc Delta function.

What is the value of DDelta_n_(c), where c is not zero?

Is this value zero?

(Here DDelta_n_(x) means the nth derivative of the Direc delta
function)...

Lucy wrote:
> Specificly, I need to find the function values of the nth derivative of
> Direc Delta function.
>
> What is the value of DDelta_n_(c), where c is not zero?
>
> Is this value zero?
>
> (Here DDelta_n_(x) means the nth derivative of the Direc delta
> function)...

Lucy,

You know that the delta function is really a distribution and as such,
one doesn't just say what is the 3rd derivative of the delta function?
The "function" only really makes sense inside of an integral.

However what you may be asking for is what if I need to find:

integral( phi(t)* nth derivative of delta(t)) dt where the interval
includes zero.

It is (-1)^n * nth derivative of phi(t) evaluated at t=0. This is found
by using tabular integration or repeated integration by parts. Of
course you can implement a simple change of variable if you wish to use
delta(t-t_0).

Is this what you need?

Clay S . Turner

Clay wrote:
> Lucy wrote:
> > Specificly, I need to find the function values of the nth derivative of
> > Direc Delta function.
> >
> > What is the value of DDelta_n_(c), where c is not zero?
> >
> > Is this value zero?
> >
> > (Here DDelta_n_(x) means the nth derivative of the Direc delta
> > function)...
>
> Lucy,
>
> You know that the delta function is really a distribution and as such,
> one doesn't just say what is the 3rd derivative of the delta function?
> The "function" only really makes sense inside of an integral.
>
> However what you may be asking for is what if I need to find:
>
> integral( phi(t)* nth derivative of delta(t)) dt where the interval
> includes zero.
>
> It is (-1)^n * nth derivative of phi(t) evaluated at t=0. This is found
> by using tabular integration or repeated integration by parts. Of
> course you can implement a simple change of variable if you wish to use
> delta(t-t_0).
>
> Is this what you need?
>
> Clay S . Turner

Hi Clay,

Thanks for the reply.

I wanted to find the value of nth derivative of delta function at any
point, because I met with the following weird thing:

We all know that cos(2*pi*t) has a Fourier transform of the following
form:

F1(f)=0.5*delta(f-1)+0.5*delta(f+1).

However, we can also expand cos(2*pi*t) as polynomials for all t on R,
using Taylor expansion.

Using the Foureir transform of polynomials(see Wiki pages of Fourier
Transform), I obtained:

F2(f)=delta(f)+1/(2!)*delta_2_(f)+1/(4!)*delta_4_(f)+1/(6!)*delta_6_(f)
+ ...

where 2! denotes the factorial of 2, delta_4_(f) denotes the 4th
derivative of delta(f) function.

Now it is very strange, how can the above two be equal?

If I can do a Taylor expansion of "delta(f+1)" and "delta(f-1", I will
make the above two expressions the same.

But can I do Taylor expansion on "delta"?

Can I use this method to evaluate the FT of some functions that
traditionally cannot be Foureir transformed(now via an approximating
polynomial method)?

Numerically, what are the values of delta's derivatives?

Lucy wrote:
(snip)

> Numerically, what are the values of delta's derivatives?

I don't know about numerically, but you can imagine that a delta
is a very sharp peak, so it goes up on one side and down on the other.

The derivative then has an up peak followed by a down peak, yet as
with delta zero on both sides.  The Nth derivative will have N+1
sharp peaks, alternating between plus and minus, and all in the
limit as their width goes to zero, and with amplitudes of the
binomial coefficients.

As Clay said, delta is only really defined in the context of
its integral, but within that if you exchange the order of
integration and the derivative you can talk about the derivatives
of the delta.

If you consider delta as the limit as the width goes to zero of
a gaussian, the derivatives will be the limit as the width goes
to zero of the Nth derivative of a gaussian.

-- glen

> Specificly, I need to find the function values of the nth derivative of
> Direc Delta function.
>
> What is the value of DDelta_n_(c), where c is not zero?
>
> Is this value zero?
>
> (Here DDelta_n_(x) means the nth derivative of the Direc delta
> function)...

See

http://en.wikipedia.org/wiki/Dirac_delta_function#Derivatives_of_the_delta_function

They are best understood by their Fourier transforms or
Laplace transforms, not as functions.

carlos@colorado.edu wrote:
> > Specificly, I need to find the function values of the nth derivative of
> > Direc Delta function.
> >
> > What is the value of DDelta_n_(c), where c is not zero?
> >
> > Is this value zero?
> >
> > (Here DDelta_n_(x) means the nth derivative of the Direc delta
> > function)...
>
> See
>
> http://en.wikipedia.org/wiki/Dirac_delta_function#Derivatives_of_the_delta_function
>
> They are best understood by their Fourier transforms or
> Laplace transforms, not as functions.


It is really sad. If I have a function that is difficult to find its
Fourier transform, I cannot expand it into polynomials and apply
Fourier transform to each of the expanded polynomials,... because that
will result in a bunch of derivative of deltas... I am very sad about
this ...

In article <1156117527.177799.141830@i42g2000cwa.googlegroups.com>,
Lucy <comtech.usa@gmail.com> wrote:
>
>carlos@colorado.edu wrote:
>> > Specificly, I need to find the function values of the nth derivative of
>> > Direc Delta function.
>> >
>> > What is the value of DDelta_n_(c), where c is not zero?
>> >
>> > Is this value zero?
>> >
>> > (Here DDelta_n_(x) means the nth derivative of the Direc delta
>> > function)...
>>
>> See
>>
>>
>http://en.wikipedia.org/wiki/Dirac_delta_function#Derivatives_of_the_delta_function
>>
>> They are best understood by their Fourier transforms or
>> Laplace transforms, not as functions.
>
>
>It is really sad. If I have a function that is difficult to find its
>Fourier transform, I cannot expand it into polynomials and apply
>Fourier transform to each of the expanded polynomials,... because that
>will result in a bunch of derivative of deltas... I am very sad about
>this ...

On the other hand, if the function is sufficiently nice you might
be able to expand it into, say, Hermite functions...

Robert Israel                                israel@math.ubc.ca
Department of Mathematics        http://www.math.ubc.ca/~israel 
University of British Columbia            Vancouver, BC, Canada

Robert Israel wrote:
> In article <1156117527.177799.141830@i42g2000cwa.googlegroups.com>,
> Lucy <comtech.usa@gmail.com> wrote:
> >
> >carlos@colorado.edu wrote:
> >> > Specificly, I need to find the function values of the nth derivative of
> >> > Direc Delta function.
> >> >
> >> > What is the value of DDelta_n_(c), where c is not zero?
> >> >
> >> > Is this value zero?
> >> >
> >> > (Here DDelta_n_(x) means the nth derivative of the Direc delta
> >> > function)...
> >>
> >> See
> >>
> >>
> >http://en.wikipedia.org/wiki/Dirac_delta_function#Derivatives_of_the_delta_function
> >>
> >> They are best understood by their Fourier transforms or
> >> Laplace transforms, not as functions.
> >
> >
> >It is really sad. If I have a function that is difficult to find its
> >Fourier transform, I cannot expand it into polynomials and apply
> >Fourier transform to each of the expanded polynomials,... because that
> >will result in a bunch of derivative of deltas... I am very sad about
> >this ...
>
> On the other hand, if the function is sufficiently nice you might
> be able to expand it into, say, Hermite functions...
>
> Robert Israel                                israel@math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada

Robert,

Thanks for your comments. Are you saying after expanding into Hermite
functions, the Hermite functions will be nice for Fourier integrals,
instead of the polynomials will only be transformed into derivatives of
deltas?

Thanks again!

Lucy wrote:
> Robert Israel wrote:
> > In article <1156117527.177799.141830@i42g2000cwa.googlegroups.com>,
> > Lucy <comtech.usa@gmail.com> wrote:
> > >
> > >carlos@colorado.edu wrote:
> > >> > Specificly, I need to find the function values of the nth derivative of
> > >> > Direc Delta function.
> > >> >
> > >> > What is the value of DDelta_n_(c), where c is not zero?
> > >> >
> > >> > Is this value zero?
> > >> >
> > >> > (Here DDelta_n_(x) means the nth derivative of the Direc delta
> > >> > function)...
> > >>
> > >> See
> > >>
> > >>
> > >http://en.wikipedia.org/wiki/Dirac_delta_function#Derivatives_of_the_delta_function
> > >>
> > >> They are best understood by their Fourier transforms or
> > >> Laplace transforms, not as functions.
> > >
> > >
> > >It is really sad. If I have a function that is difficult to find its
> > >Fourier transform, I cannot expand it into polynomials and apply
> > >Fourier transform to each of the expanded polynomials,... because that
> > >will result in a bunch of derivative of deltas... I am very sad about
> > >this ...
> >
> > On the other hand, if the function is sufficiently nice you might
> > be able to expand it into, say, Hermite functions...
> >
> > Robert Israel                                israel@math.ubc.ca
> > Department of Mathematics        http://www.math.ubc.ca/~israel
> > University of British Columbia            Vancouver, BC, Canada
>
> Robert,
>
> Thanks for your comments. Are you saying after expanding into Hermite
> functions, the Hermite functions will be nice for Fourier integrals,
> instead of the polynomials will only be transformed into derivatives of
> deltas?

That's right.  The Hermite functions are eigenfunctions of the Fourier
transform.  See e.g. <http://en.wikipedia.org/wiki/Hermite_polynomials>

 Robert Israel                                israel@math.ubc.ca
 Department of Mathematics        http://www.math.ubc.ca/~israel
 University of British Columbia            Vancouver, BC, Canada

Lucy wrote:

(snip)
> It is really sad. If I have a function that is difficult to find its
> Fourier transform, I cannot expand it into polynomials and apply
> Fourier transform to each of the expanded polynomials,... because that
> will result in a bunch of derivative of deltas... I am very sad about
> this ...

Various identities involving delta are shown in:

http://mathworld.wolfram.com/DeltaFunction.html

-- glen