Hi!(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to calculate undulations for a smectic A liquid crystal for 1,..,4 dimensions. The general equation for [itex]d[/itex] dimensions are

[itex]\langle u^2(\mathbf{x})\rangle = \frac{k_BT}{(2\pi)^dB} \int_\frac{1}{L}^{q_c}\frac{\text{d}q_\parallel \text{d}^dq_\perp}{q_\parallel+\lambda^2q_\perp^4}

[/itex]

my problem (so far) is for 2 dimensions:

approximately the problem reduces to

[itex]

\langle u^2(\mathbf{x})\rangle = \frac{k_BT}{(2\pi)^dB} \int_\frac{1}{L}^{q_c}\text{d}q_\perp\, \arctan\left(\frac{q_c}{\lambda q_\perp^2}\right)\frac{1}{\lambda q_\perp^2}

[/itex]

or, eventually, is there a more clever way to use the first equation? I have tried to use subs with the arctan function and partial integration.

With help from Abramowitz' Handbook of Mathical Function the second equation can be written as

[itex]

\langle u^2(\mathbf{x})\rangle =-\frac{\lambda}{q_c}\left(\frac{q_c}{\lambda}\right)^{3/2} \frac{k_BT}{(2\pi)^dB} \left[2\sqrt{q_\perp} \arctan{q_c} - 2\int\frac{\sqrt{q_\perp}\text{d}q_\perp}{1+ q_\perp^2}\right]

[/itex]

but I think I am moving in circles now.. so my problem is basically to calculate either

[itex]

\int\text{d}x\arctan(a/x^2)\frac{1}{x^2}

[/itex]

or

[itex]

\int\frac{\text{d}x \sqrt{x}}{1+x^2}

[/itex]

will be grateful for some help! :)

Al

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Integration of arctan

Loading...

Similar Threads - Integration arctan | Date |
---|---|

Integrate arctan(y) dx | Jan 27, 2011 |

Integrate y= arctan x for 0< x <1 | Nov 28, 2010 |

Arctan integral | Jul 8, 2008 |

Integration using u substitution and arctan | Mar 25, 2008 |

Particular Integral of arctan example | Aug 17, 2004 |

**Physics Forums - The Fusion of Science and Community**