# Find Möbius transformation that send imaginary axis to a circle and the real axis to itself

Find a linear fractional transformation $T$ that maps the real axis onto itself and the imaginary axis onto the circle $|w+\frac{5}{4}|=\frac{3}{4}.$

I just have no clue to do it. can i get some help?

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What have you tried so far? – Loki Clock Apr 16 '13 at 0:08
en.wikipedia.org/wiki/Cross-ratio (Hint: A LFT is determined by the images of 0, 1, and $\infty$.) – Daenerys Naharis Apr 16 '13 at 0:45
@Loki Clock i just have no clue how to do it.. should i use the cross ratio or something first? – Zhongyuan Liu Apr 16 '13 at 2:51
The first thing is to find out how the mapping behaves on the real and imaginary lines as the parameters change. Draw lines or make figures that point out what you observe, then relate the quantities of the measuring system to special points like the values of i and 1 to inquire about the validity of your observations. – Loki Clock Apr 17 '13 at 2:58

It may be easier to find the inverse transformation. The circle meets the real axis at the points $w=-2$ and $w=-1/2$. These points must go to the intersection points of the real and imaginary axes, which are $0$ and $\infty$. The map $z=\dfrac{w+1/2}{w+2}$ does the job. The last step is to find the inverse: solve for $w$ in terms of $z$.