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

The following function [itex]\Omega = [x^2 + y^2 + (z - ia)^2]^{1/2}[/itex] has the following branch points:

[itex] z_1 = i (a + \lambda), z_2 = i (a - \lambda) [/itex] where [itex] \lambda^2 = x^2 + y^2 [/itex].

Now I do not understand the theory about branch cuts. It is said in the paper I read: "We make cuts on the imaginary z axis from [itex] z_1 [/itex] up to [itex] z = i\infty [/itex] and from [itex] z_2 [/itex] down to [itex] z = -i\infty [/itex]". OK. Now, let's consider the case [itex] \lambda < a [/itex]. Then the lower branch line crosses the real z axis. What I do not understand is that

[itex]\Omega^- = -\Omega^+ , z=0 [/itex]

where [itex]\Omega^- [/itex] denotes the value of the radical for [itex] z = 0^- [/itex] and [itex]\Omega^+ [/itex] for [itex] z = 0^+ [/itex]. (By the way, what is 'radical'?) They further wrote that

[itex] \Omega^+ = [\lambda^2 + (-ia)^2]^{1/2} = -i(a^2-\lambda^2)^{1/2} [/itex]. How they calculated that? When I tried to calculate that I got +i instead of -i for the prefactor.

I thought that that lower branch cut means that the angle is from the interval [itex] (-\pi/2,3/2\pi) [/itex] and so

[itex] \Omega^+ = [\lambda^2 + (\epsilon -ia)^2]^{1/2} = [\lambda^2 -a^2 + \epsilon^2 - 2\epsilon ai]^{1/2} = [a^2 - \lambda^2]^{1/2} e^{i\pi/2} = [a^2 - \lambda^2]^{1/2} i[/itex] which is wrong.

Any help much appreciated.

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

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!

# Branch points and branch cuts

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Branch points branch | Date |
---|---|

I Domain of single-valued logarithm of complex number z | Nov 28, 2016 |

Distribution between non-linear branches | Jun 9, 2015 |

How to derive the branch cuts for complex arcsin(z) | Oct 8, 2014 |

What branches of mathematics do I need to know to make AI algorithms? | Sep 27, 2014 |

Calculating the intersection of a branching point | Nov 14, 2012 |

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