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In his book on Complex Analysis, Ahlfors on Page $17$ writes:

  • "A directed line $z = a + bt$ determines a right half plane consisting of all points $z$ with $\Im \left(\frac{z - a}b\right) < 0$ and a left half plane with $\Im \left(\frac{z - a}b\right) > 0$"

Why does Ahlfors use $\Im$ with $\frac{z - a}b$?

I thought $t = \frac{z - a}b$ belongs to Reals. So $\frac{z - a}b>0$ would represents the Right Half and $\frac{z - a}b<0$ the Left Half of the plane.

Definitely I'm missing something since Ahlfors is using Im with $\frac{z - a}b$.

user
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sidharth
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1 Answers1

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The operation

$$z \to \frac {z-a}b$$

corresponds to a translation of $-a$ (that is $a$ goes at the origin) and a rotation of an angle equal to $\theta =-\arg (b)$ in such way that the line $z=a+bt$ goes onto the real axis and therefore we can conclude according to the statement given by Ahlfors.

user
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  • Thanks. If we are mapping onto the Real Line, why do we use Right Hand and Left Hand Plane and not Upper Half and Lower Half. Apologies if the question is a bit silly. – sidharth May 12 '23 at 14:22
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    @sidharth It's a good question. I think that the terminology used by Ahlfors refers to an observer moving along the positive direction of the "directed line". In this sense the points with $\Im(z)>0$ lie on the left hand. – user May 12 '23 at 14:50
  • Thanks!! Very Helpful – sidharth May 13 '23 at 04:31
  • Also can you suggest a book as rigourous as Ahlfors but where I dont have to struggle so much with explanations. Am learning maths on my own. – sidharth May 13 '23 at 04:38
  • You can refer to https://math.stackexchange.com/q/30749/505767 – user May 13 '23 at 08:39
  • Its a long list :) Thanks anyways. Your explanations were really helpful. Dont think I would have understood on my own – sidharth May 13 '23 at 12:13
  • You are welcome! Bye – user May 13 '23 at 13:58