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[SOLVED] fractional linear transformation--conformal mapping

dwsmith

Well-known member
Feb 1, 2012
1,673
Find necessary and sufficient conditions on the real numbers $a$, $b$, $c$, and $d$ such that the fractional linear transformation
$$
f(z) = \frac{az + b}{cz + d}
$$
maps the upper half plane to itself.

I just need some guidance on starting this one since I am not sure on how to begin.
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,707
Find necessary and sufficient conditions on the real numbers $a$, $b$, $c$, and $d$ such that the fractional linear transformation
$$
f(z) = \frac{az + b}{cz + d}
$$
maps the upper half plane to itself.

I just need some guidance on starting this one since I am not sure on how to begin.
Begin by multiplying top and bottom by the complex conjugate of the denominator: $$f(z) = \dfrac{az + b}{cz + d} = \dfrac{(az + b)(c\overline{z} + d)}{(cz + d)(c\overline{z} + d)}.$$

The denominator in that last fraction is real, so you need to find the imaginary part of the numerator and investigate what makes it positive whenever $z$ has positive imaginary part.
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Begin by multiplying top and bottom by the complex conjugate of the denominator: $$f(z) = \dfrac{az + b}{cz + d} = \dfrac{(az + b)(c\overline{z} + d)}{(cz + d)(c\overline{z} + d)}.$$

The denominator in that last fraction is real, so you need to find the imaginary part of the numerator and investigate what makes it positive whenever $z$ has positive imaginary part.
So $ad > bc$
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,707