# Need a solution for the following problem

1. Jan 27, 2007

### Showstopper

The question is attached. Thanks a lot

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2. Jan 27, 2007

### arildno

Expressed as an integral, what is F(0) equal to?

3. Jan 27, 2007

### Showstopper

F(0) = Fo where Fo can be any constant, but we have to specify it.

4. Jan 27, 2007

### Showstopper

not specify but determine actually my bad

5. Jan 27, 2007

### arildno

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

6. Jan 27, 2007

### Showstopper

see attached

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• ###### F(0).JPG
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7. Jan 27, 2007

### arildno

Indeed!
And what is the exact value of that integral?

8. Jan 27, 2007

### Showstopper

- e of positive infinity??

9. Jan 27, 2007

### arildno

Okay, so you are unfamiliar with the famous result:
$$\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}$$

10. Jan 27, 2007

### Showstopper

yea, never seen this before

11. Jan 27, 2007

### arildno

Okay, so now you know the value of F(0)!

12. Jan 27, 2007

Care to walk me through?

13. Jan 28, 2007

So what do we do with this?
Sorry, im not understanding.

14. Jan 28, 2007

### HallsofIvy

Staff Emeritus
The problem SAID "Write a differential equation for F(x):
$$\frac{dF}{d\omega}+ h(\omega)F$$
with F(0)= F0 where F0 is the constant you have to determine explicitly."

Okay, you now know that F0= $\sqrt{\pi}$.

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
$$\frac{dF}{d\omega}= h(\omega)F$$
or
$$\frac{dF}{d\omega}+ h(\omega)F= 0$$

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

15. Jan 28, 2007

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

is it that straight forward? Am I not understanding something?

16. Jan 28, 2007

### Showstopper

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

17. Jan 29, 2007

### HallsofIvy

Staff Emeritus
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 $h(\omega)$ so that F(x), as given in integral form, satisfies that equation.

What happens if you differentiate the integral defining F?

18. Jan 29, 2007

I got F(w) = root( pi) when solving the diff. equation

So is this the answer to the integral?

19. Jan 30, 2007

### HallsofIvy

Staff Emeritus
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
$$F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx$$
What do you get if you differentiate that equation with respect to $\omega$? (In this case it is legitimate to simplydifferentiate inside the integral.)

By the way, $$e^{i\omega x}e^{-x^2}= e^{i\omega x- x^2}$$

20. Jan 30, 2007

The derivative of $$F(\omega)= \int_{-\infty}^{\infty}e^{-\omega x}e^{-x^2}dx$$ with respect to $\omega$ is $$F(\omega)= \int_{-\infty}^{\infty}-xe^{-\omega x}e^{-x^2}dx$$