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 $b$.