 Home

Codex Wiki

List of universities

University of Oxford

Course of the University of Oxford

Undergraduate course of the University of Oxford

Physics course of the University of Oxford

Short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford

2015

Take $g(z)=L(z)^3/z^3$ on the same <logarithm branch>. At zero, $L(z)^3=-z^3+O(z^4)$, so the apparent singularity is removable and the closed <contour integral> is zero. The bank difference is
$$
(\ell-i\pi)^3-(\ell+i\pi)^3=-6i\pi\ell^2+2i\pi^3.
$$
The circular contributions vanish by the previous estimates with the third power of the logarithm. Consequently
$$
0=-6i\pi\int_1^\infty\frac{\log^2(x-1)}{x^3}\,dx+2i\pi^3\int_1^\infty\frac{dx}{x^3}.
$$
The last <integral> is $1/2$, giving
$$
\boxed{\int_1^\infty\frac{\log^2(x-1)}{x^3}\,dx=\frac{\pi^2}{6}.}
$$
Increasing the logarithm power has converted the desired real <integral> into a jump across a <branch cut>, while removing the pole inside the contour.

Solution

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S1 / 1 / 3 / Solution

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S1 / 1 / 3 / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S1 / 1 / 3 / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)