Circle in the Complex Domain where Mean is not the Centre

[electronicengi]

Hello people of Physics Forums,

In my research into transmission lines, I have come across the following function:

x = ( a - i * b * tan(t) ) / ( c - i * d * tan(t) )

In the above equation x, a, b, c and d are complex and t is real. If my analysis is correct, varying t from -pi/2 to pi/2 will yield a circle in the complex domain that intersects the points a/c and b/d.

I would like to know more about this type of function. Has it been studied before? If so, does it have some sort of special name that I can look up in a mathematics textbook to learn more about it? In particular, I am interested in finding the "average" value of x; does a closed form solution (in terms of a, b, c and d) exist if one integrates x from t = -pi/2 to pi/2?

Thank you in advance.

electronicengi

 

[micromass]

Well, you are basically interested in $t\rightarrow T(-i\textrm{tan}(t))$ where $T$ is a Mobius transformation.

See http://en.wikipedia.org/wiki/Mobius_transformation

The image is indeed a circle (well, generalized circle), which is a simple consequence of fact that Mobius transformations send circles to circles: http://en.wikipedia.org/wiki/Mobius_transformation#Preservation_of_angles_and_generalized_circles

 

[electronicengi]

Excellent. Looks like I will be doing a little bit of reading up on the Mobius transformation.

Does anyone know how to find (if possible) a closed form solution for the mean value of x?

 

[micromass]

Excellent. Looks like I will be doing a little bit of reading up on the Mobius transformation.

Does anyone know how to find (if possible) a closed form solution for the mean value of x?

No, but I think the following should be true:
Let $a = T(0)$, let $b = T(i\pi/2) + T(-i\pi/2)$. Then the mean value lies on the line through $a$ and $b$.

 

[blue_raver22]

Hello electronicengi,

Thank you for sharing your findings with us. It is interesting to see a circle in the complex domain where the mean is not the center. This function does not seem to have a specific name, but it falls under the category of rational functions in the complex domain.

In terms of previous studies, there has been research on the behavior of complex rational functions and their geometric properties. Some studies have also looked at the behavior of these functions on the Riemann sphere, which is a one-point compactification of the complex plane.

As for finding the average value of x, it is possible to calculate it by integrating the function over the given range. However, it may not have a closed-form solution in terms of a, b, c, and d. This would depend on the specific values of these parameters and the properties of the function.

I suggest looking into books on complex analysis and rational functions for more information on this topic. It is an interesting area of study and I hope your research leads to further insights. Good luck!