Integrate $int_{-infty}^infty [4(log r_1 - log r_2) - 2(x_1^2/r_1^2 - x_2^2/r_2^2)]^2 dx$
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As the title suggests, I am having trouble evaluating the following definite integral:
$$int_{-infty}^infty left[4left(log r_1 - log r_2right) - 2left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right)right]^2 dx$$
where
$$begin{align}
x_1 &= x-a\
x_2 &= x+a\
r_1^2 &= x_1^2 + z^2\
r_2^2 &= x_2^2 + z^2
end{align}$$
and $a > 0$, $z > 0$.
I've started by expanding the square, which gives
$$int_{-infty}^infty left[16left(log r_1 - log r_2right)^2 - 16left(log r_1 - log r_2right)left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right) + 4left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right)^2right] dx.$$
I managed to integrate the rightmost term $4(x_1^2/r_1^2 - x_2^2/r_2^2)^2$ using a partial fraction decomposition. However, I'm struggling with the two remaining terms: $16(log r_1 - log r_2)^2$ and $-16(log r_1 - log r_2)(x_1^2/r_1^2 - x_2^2/r_2^2)$.
I only know that the integral converges, as I am able to approach it numericaly by fixing $a$ and $z$.
If anyone has an idea on how to proceed, or, even better, has a solution, I'll take it.
Thanks!
calculus integration logarithms polylogarithm
asked Nov 21 at 15:15
Son Pham-Ba
61
add a comment |
up vote
1
down vote
favorite
As the title suggests, I am having trouble evaluating the following definite integral:
$$int_{-infty}^infty left[4left(log r_1 - log r_2right) - 2left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right)right]^2 dx$$
where
$$begin{align}
x_1 &= x-a\
x_2 &= x+a\
r_1^2 &= x_1^2 + z^2\
r_2^2 &= x_2^2 + z^2
end{align}$$
and $a > 0$, $z > 0$.
I've started by expanding the square, which gives
$$int_{-infty}^infty left[16left(log r_1 - log r_2right)^2 - 16left(log r_1 - log r_2right)left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right) + 4left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right)^2right] dx.$$
I managed to integrate the rightmost term $4(x_1^2/r_1^2 - x_2^2/r_2^2)^2$ using a partial fraction decomposition. However, I'm struggling with the two remaining terms: $16(log r_1 - log r_2)^2$ and $-16(log r_1 - log r_2)(x_1^2/r_1^2 - x_2^2/r_2^2)$.
I only know that the integral converges, as I am able to approach it numericaly by fixing $a$ and $z$.
If anyone has an idea on how to proceed, or, even better, has a solution, I'll take it.
Thanks!
calculus integration logarithms polylogarithm
asked Nov 21 at 15:15
Son Pham-Ba
61
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
As the title suggests, I am having trouble evaluating the following definite integral:
$$int_{-infty}^infty left[4left(log r_1 - log r_2right) - 2left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right)right]^2 dx$$
where
$$begin{align}
x_1 &= x-a\
x_2 &= x+a\
r_1^2 &= x_1^2 + z^2\
r_2^2 &= x_2^2 + z^2
end{align}$$
and $a > 0$, $z > 0$.
I've started by expanding the square, which gives
$$int_{-infty}^infty left[16left(log r_1 - log r_2right)^2 - 16left(log r_1 - log r_2right)left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right) + 4left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right)^2right] dx.$$
I managed to integrate the rightmost term $4(x_1^2/r_1^2 - x_2^2/r_2^2)^2$ using a partial fraction decomposition. However, I'm struggling with the two remaining terms: $16(log r_1 - log r_2)^2$ and $-16(log r_1 - log r_2)(x_1^2/r_1^2 - x_2^2/r_2^2)$.
I only know that the integral converges, as I am able to approach it numericaly by fixing $a$ and $z$.
If anyone has an idea on how to proceed, or, even better, has a solution, I'll take it.
Thanks!
calculus integration logarithms polylogarithm
asked Nov 21 at 15:15
Son Pham-Ba
61
As the title suggests, I am having trouble evaluating the following definite integral:
$$int_{-infty}^infty left[4left(log r_1 - log r_2right) - 2left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right)right]^2 dx$$
where
$$begin{align}
x_1 &= x-a\
x_2 &= x+a\
r_1^2 &= x_1^2 + z^2\
r_2^2 &= x_2^2 + z^2
end{align}$$
and $a > 0$, $z > 0$.
I've started by expanding the square, which gives
$$int_{-infty}^infty left[16left(log r_1 - log r_2right)^2 - 16left(log r_1 - log r_2right)left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right) + 4left(frac{x_1^2}{r_1^2} - frac{x_2^2}{r_2^2}right)^2right] dx.$$
I managed to integrate the rightmost term $4(x_1^2/r_1^2 - x_2^2/r_2^2)^2$ using a partial fraction decomposition. However, I'm struggling with the two remaining terms: $16(log r_1 - log r_2)^2$ and $-16(log r_1 - log r_2)(x_1^2/r_1^2 - x_2^2/r_2^2)$.
I only know that the integral converges, as I am able to approach it numericaly by fixing $a$ and $z$.
If anyone has an idea on how to proceed, or, even better, has a solution, I'll take it.
Thanks!
calculus integration logarithms polylogarithm
calculus integration logarithms polylogarithm
asked Nov 21 at 15:15
Son Pham-Ba
61
asked Nov 21 at 15:15
Son Pham-Ba
61
asked Nov 21 at 15:15
Son Pham-Ba
61
asked Nov 21 at 15:15
Son Pham-Ba
61
asked Nov 21 at 15:15
Son Pham-Ba
61
61
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add a comment |
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