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 Post subject: Lim and integration Delta functionPosted: Fri, 8 Jul 2011 17:09:08 UTC
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Joined: Fri, 8 Jul 2011 16:38:49 UTC
Posts: 10
1.
2. Find Delta function :

Thank alot.

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 Post subject: Re: Lim and integration Delta functionPosted: Fri, 8 Jul 2011 19:05:13 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
Ipie wrote:
1.
2. Find Delta function :

Thank alot.

Um, that first one makes no sense, there is no dependence on beta with that limit there.

The second one, what do you mean "find Delta function"? Are you supposed to define a function such that it satisfies that integral equation?

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 Post subject: Re: Lim and integration Delta functionPosted: Sat, 9 Jul 2011 00:57:50 UTC
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Joined: Fri, 8 Jul 2011 16:38:49 UTC
Posts: 10
Ipie wrote:
1.
2. Find Delta function :

Thank alot.

Um, that first one makes no sense, there is no dependence on beta with that limit there.

The second one, what do you mean "find Delta function"? Are you supposed to define a function such that it satisfies that integral equation?

First one is a kind of Gaussian function. When beta becomes very high that function approaches Delta function.
'Delta function is a generalized function that can be defined as the limit of class delta sequences'-Mathworld. and the functional nature of the delta function is often suppressed.
Delta function has fundamental property:

So I need to find delta function

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 Post subject: Re: Lim and integration Delta functionPosted: Sat, 9 Jul 2011 04:26:37 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Ipie wrote:
Ipie wrote:
1.
2. Find Delta function :

Thank alot.

Um, that first one makes no sense, there is no dependence on beta with that limit there.

The second one, what do you mean "find Delta function"? Are you supposed to define a function such that it satisfies that integral equation?

First one is a kind of Gaussian function. When beta becomes very high that function approaches Delta function.
'Delta function is a generalized function that can be defined as the limit of class delta sequences'-Mathworld. and the functional nature of the delta function is often suppressed.
Delta function has fundamental property:

So I need to find delta function

Delta function isn't a function, but a measure. What exactly are you trying to find here? If you are trying to find the measure, then it is simply

for any set , so

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 Post subject: Posted: Wed, 13 Jul 2011 07:31:31 UTC
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Joined: Fri, 8 Jul 2011 16:38:49 UTC
Posts: 10
Thanks to @outermeasure. Do you know about Delta function applications?

with initial value x(0)=3

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 Post subject: Re: Lim and integration Delta functionPosted: Wed, 13 Jul 2011 08:57:13 UTC
 Member

Joined: Fri, 8 Jul 2011 16:38:49 UTC
Posts: 10
Ipie wrote:
1.
2. Find Delta function :

Thank alot.

Um, that first one makes no sense, there is no dependence on beta with that limit there.

The first one:

Apply Taylor series expansion for

So

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 Post subject: Posted: Wed, 13 Jul 2011 09:59:10 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Ipie wrote:
Thanks to @outermeasure. Do you know about Delta function applications?

with initial value x(0)=3

You have (where the derivative is the weak derivative), so

(remember you can change the value on a Lebesgue null set without affecting the weak derivative).

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 Post subject: Re: Lim and integration Delta functionPosted: Wed, 13 Jul 2011 10:00:48 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Ipie wrote:
Ipie wrote:
1.
2. Find Delta function :

Thank alot.

Um, that first one makes no sense, there is no dependence on beta with that limit there.

The first one:

Apply Taylor series expansion for

So

It still doesn't make any sense --- you have committed the cardinal sin of using the same variable for two different things. Which is from the and which is your dummy ?

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 Post subject: Re: Lim and integration Delta functionPosted: Wed, 13 Jul 2011 12:32:10 UTC
 Member

Joined: Fri, 8 Jul 2011 16:38:49 UTC
Posts: 10
outermeasure wrote:
Ipie wrote:
Ipie wrote:
1.
2. Find Delta function :

Thank alot.

Um, that first one makes no sense, there is no dependence on beta with that limit there.

The first one:

Apply Taylor series expansion for

So

It still doesn't make any sense --- you have committed the cardinal sin of using the same variable for two different things. Which is from the and which is your dummy ?

I am very sorry. It's only f, not . So it should be:

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 Post subject: Posted: Wed, 13 Jul 2011 15:39:34 UTC
 Member

Joined: Fri, 8 Jul 2011 16:38:49 UTC
Posts: 10
outermeasure wrote:
Ipie wrote:
Thanks to @outermeasure. Do you know about Delta function applications?
(1)

with initial value x(0)=3

You have (where the derivative is the weak derivative), so

(remember you can change the value on a Lebesgue null set without affecting the weak derivative).

The general solution of (1) is:

where,

or

and

so (2)

With the initial condition x(0)=3, we get:

So c = 3

Also from the delta function definition:

Substitute into (2)m yields,

Is this procedure OK?

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