Need a solution for the following problem

The question is attached. Thanks a lot
 

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arildno

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Expressed as an integral, what is F(0) equal to?
 
F(0) = Fo where Fo can be any constant, but we have to specify it.
 
not specify but determine actually my bad
 

arildno

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I SAID:
Expressed as an integral, WHAT IS F(0)?
 
see attached
 

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arildno

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Indeed!
And what is the exact value of that integral?
 
- e of positive infinity??
 

arildno

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Okay, so you are unfamiliar with the famous result:
[tex]\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}[/tex]
 
yea, never seen this before
 

arildno

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Okay, so now you know the value of F(0)! :smile:
 
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So what is the answer?
Care to walk me through?
 

HallsofIvy

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So what do we do with this?
Sorry, im not understanding.
The problem SAID "Write a differential equation for F(x):
[tex]\frac{dF}{d\omega}+ h(\omega)F[/tex]
with F(0)= F0 where F0 is the constant you have to determine explicitly."

Okay, you now know that F0= [itex]\sqrt{\pi}[/itex].

By the way, I notice that the "differential equation" you give (I have copied it exactly above) is not an equation! There is no equal sign. I imagine it is actually either
[tex]\frac{dF}{d\omega}= h(\omega)F[/tex]
or
[tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex]

They differ, of course, only in the sign of h.
 
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The problem SAID "Write a differential equation for F(x):
[tex]\frac{dF}{d\omega}+ h(\omega)F[/tex]
with F(0)= F0 where F0 is the constant you have to determine explicitly."

Okay, you now know that F0= [itex]\sqrt{\pi}[/itex].

By the way, I notice that the "differential equation" you give (I have copied it exactly above) is not an equation! There is no equal sign. I imagine it is actually either
[tex]\frac{dF}{d\omega}= h(\omega)F[/tex]
or
[tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex]

They differ, of course, only in the sign of h.

Oh so we just have to solve the ODE [tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex] with initial condition F0= [itex]\sqrt{\pi}[/itex]...

is it that straight forward? Am I not understanding something?
 
but how do you solve a question like this, with 3 variables and a complex number? even if we use the method of solving linear differential equations
 

HallsofIvy

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Oh so we just have to solve the ODE [tex]\frac{dF}{d\omega}+ h(\omega)F= 0[/tex] with initial condition F0= [itex]\sqrt{\pi}[/itex]...

is it that straight forward? Am I not understanding something?
Apparently you are having trouble reading the problem- which I just quoted.

The problem does NOT ask you to solve any differential equation. It asks you to FIND an equation of that form (essentially find the function [itex]h(\omega)[/itex] so that F(x), as given in integral form, satisfies that equation.

What happens if you differentiate the integral defining F?
 
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I got F(w) = root( pi) when solving the diff. equation

So is this the answer to the integral?
 

HallsofIvy

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This is becoming very frustrating. Is there any point in responding if you don't read the replies?? I just said, the problem does not ask you to solve a differential equation, it asks you to find a differential equation for F! And I don't know what you mean by "the answer to this integral". Integrals don't have answers, questions do. What is the question?

You are told that F is defined by
[tex]F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx[/tex]
What do you get if you differentiate that equation with respect to [itex]\omega[/itex]? (In this case it is legitimate to simplydifferentiate inside the integral.)

By the way, [tex]e^{i\omega x}e^{-x^2}= e^{i\omega x- x^2}[/tex]
 
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This is becoming very frustrating. Is there any point in responding if you don't read the replies?? I just said, the problem does not ask you to solve a differential equation, it asks you to find a differential equation for F! And I don't know what you mean by "the answer to this integral". Integrals don't have answers, questions do. What is the question?

You are told that F is defined by
[tex]F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx[/tex]
What do you get if you differentiate that equation with respect to [itex]\omega[/itex]? (In this case it is legitimate to simplydifferentiate inside the integral.)

By the way, [tex]e^{i\omega x}e^{-x^2}= e^{i\omega x- x^2}[/tex]
I applogize, it's not every day I come accross questions like this...this is very challenging.

I got confuzed between "solving the differential equation" and "finding the differenatial equation of F" -- I don't know the difference. :uhh:

The derivative of [tex]F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx[/tex] with respect to [itex]\omega[/itex] is [tex]F(\omega)= \int_{-\infty}^{\infty}-xe^{-\omega x}e^{-x^2}dx[/tex]
 
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Stupid latex...that's not the integral...latex keeps giving the wrong one
 

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