How to calculate this integral?

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In summary: since each q^2+q\cdot k+q\cdot(1,1,1) is a triple integral, we can just use:\mathop\iint\limits_{k_x^2+k_y^2+k_z^2>1}\iiint_{(q_x-k_x)^2+(q_y-k_y)^2+(q_z-k_z)^2\leq 1} \exp\{-(q^2+q\cdot k+q\cdot(1,1,1))\}d^3q d^3kand that will give us
  • #1
kassem84
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Hello,
I have some difficulties of calculating the following integral:
[itex] I=\int _{D}\:\:\:d^{3}q\: d^{3}k\: d^{3}p\:\:F(q^{2}, q.k, q.p, k^{2}, p^{2})[/itex]
where:
D=|k|>1, |k+q|<1 and |p-q|<1

Thanks in advance.
 
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  • #2
What is the function F?
 
  • #3
muppet said:
What is the function F?
Hello,
F=e[itex]^{-(q^{2}+q.k+q.p)}[/itex]
The most important thing is how to obtain the boundaries of the integrals. i.e. q,p,k go from where to where?
Thanks.
 
  • #4
I'm not sure I'm interpreting your notation correctly. Is that dot products in there? Also, the d^3 notation. Is that vectors? Tell you what, if it was just:

[tex]\int\int\int f(q,k,p) dqdkdp[/tex]

then I think we can use Mathematica to obtain the boundaries.
 
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  • #5
jackmell said:
I'm not sure I'm interpreting your notation correctly. Is that dot products in there? Also, the d^3 notation. Is that vectors? Tell you what, if it was just:

[tex]\int\int\int f(q,k,p) dqdkdp[/tex]

then I think we can use Mathematica to obtain the boundaries.

Yes, it is the correct notation of the integral I. All the vectors are 3-dimensional in the definition of the function and in the boundary D.
Thanks.
 
  • #6
The scalar version is quite interesting. There are two rhomboid regions to integrate over since |k|>1. I believe this is the integral for the region k>1:


[tex]\mathop\iiint\limits_{D} f(p,q,k)dpdqdk=\int_{1}^{\infty}\int_{-1-k}^{1-k}\int_{q-k}^{q+k} f(p,q,k)dpdqdk[/tex]


Perhaps the vector version is similar and you can adapt it to this one.
 
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  • #7
jackmell said:
I'm not sure I'm interpreting your notation correctly. Is that dot products in there? Also, the d^3 notation. Is that vectors? Tell you what, if it was just:

[tex]\int\int\int f(q,k,p) dqdkdp[/tex]

then I think we can use Mathematica to obtain the boundaries.

How can we use mathematica to determine the boundaries- the intersection of the three spheres?
 
  • #8
jackmell said:
The scalar version is quite interesting. There are two rhomboid regions to integrate over since |k|>1. I believe this is the integral for the region k>1:


[tex]\mathop\iiint\limits_{D} f(p,q,k)dpdqdk=\int_{1}^{\infty}\int_{-1-k}^{1-k}\int_{q-k}^{q+k} f(p,q,k)dpdqdk[/tex]


Perhaps the vector version is similar and you can adapt it to this one.

Thanks. For the vector version, it difficult for me to determine the boundaries on the angles θ and [itex]\phi[/itex].
 
  • #9
Could you or someone else tell me if I'm interpreting this correctly since I've never worked on one like this before. But first, let's just restrict it to a double integral for now:

[tex]\mathop\iint\limits_{D} f(k,q)d^3q\, d^3 k[/tex]

where each integral is a triple integral in spherical coordinates and:
D={|k|>1, |k+q|<1}

We can compute the outer one easily. Since |k|>1, then for spherical coordinate r, we can write:

[tex]\mathop\int_{r>1}\left( \mathop\int\limits_{S} f(k,q) d^3 q\right)\,d^3 k[/tex]

So what is S? Since |k+q|<1, then that means we need:

[tex]\sqrt{(k_x+q_z)^2+(k_y+q_y)^2+(k_z+q_z)^2}<1[/tex]

for every point in k-space (k_x, k_y, k_z). Now suppose we have for a particular point:

[tex]k=(3,4,7)[/tex]

Then for |k+q|<1, we would have to integrate in q-space over a sphere centered at q=(-3,-4,-7) with radius one. The boundary for that one k-point would be:

[tex](q_x+3)^2+(q_y+4)^2+(q_z+7)^2=1[/tex]

So for just that one k-point, the integral would be:

[tex]\mathop\iiint\limits_{(q_x+3)^2+(q_y+4)^2+(q_z+7)^2\leq 1} f(k,q)d^3q[/tex]

and therefore for all of the k-space, we could then write:

[tex]\mathop\iiint\limits_{k_x^2+k_y^2+k_z^2>1}\iiint_{(q_x-k_x)^2+(q_y-k_y)^2+(q_z-k_z)^2\leq 1} f(k,q)d^3q d^3k[/tex]

Ok, so just for now, can we let p be what ever it has to be to work, say p=(1,1,1) or whatever, can we now compute:

[tex]\mathop\iiint\limits_{k_x^2+k_y^2+k_z^2>1}\iiint_{(q_x-k_x)^2+(q_y-k_y)^2+(q_z-k_z)^2\leq 1} \exp\{-(q^2+q\cdot k+q\cdot(1,1,1))\}d^3q d^3k[/tex]
 
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1. How do I know which method to use for solving an integral?

There are several methods for solving integrals, including substitution, integration by parts, and trigonometric identities. The best method to use will depend on the form of the integral and the techniques you are comfortable with. It is always a good idea to try multiple methods and see which one gives the most straightforward solution.

2. What are the steps for solving an integral?

The general steps for solving an integral are:

  1. Simplify the integrand as much as possible
  2. Determine the limits of integration (if not given)
  3. Choose a method for solving the integral
  4. Apply the chosen method to solve the integral
  5. Check your answer for accuracy

3. Can I use a calculator to solve integrals?

Most scientific calculators have a built-in integral function, which can be used to solve simple integrals. However, for more complex integrals, it is recommended to use analytical methods to ensure accuracy.

4. Is there a shortcut for solving integrals?

There is no one-size-fits-all shortcut for solving integrals. However, there are some common techniques and tricks that can be used to simplify integrals, such as using trigonometric identities, substitution, and integration by parts. It is important to practice and become familiar with these techniques to effectively solve integrals.

5. How do I know if my answer to an integral is correct?

To check the accuracy of your answer, you can differentiate the result and see if it matches the original integrand. You can also use online calculators or software to verify your answer. If you are unsure, it is always a good idea to double-check your work or seek assistance from a tutor or colleague.

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