Conformal map example $ f(z)=e^z$

Question

I an studying the example-1.

I understand $f(z)=e^z$ has a nonzero derivative at all points, hence it is everywhere conformal and locally $1-1$.

But I dont understand th part I underlined with yellow pencil. Please explain it. Thank you so much:)

Answer 1

$z=x+iy$, $f(z)=e^z$, so $e^{c+iy}=e^c(\cos y+i\sin y)$ assuming $x=c$ constant is a circle with radius $e^c$

and $e^{x+ic}=e^x(\cos c+i\sin c)$

The Modulus of $f(z) = e^x,$ argument $f(z) = y.$

Horizontal lines in the cartesian plane have a constant $y$ value so the argument of the map of the horizontal line is also constant while the norm changes with $e^x$ so horizontal lines are mapped to rays with argument $y.$

vertical lines have a constant $x$ component so the map of vertical lines have a constant modulus so vertical lines get mapped to circles centered at the origin with radius $e^x$

Answer 2

Conformal maps preserve angles, it is their defining property. From wikipedia, "a conformal map is a function which preserves angles". So when an angle is mapped from the pre-image to the image, it is the same if the map is conformal at that point. If it is not conformal, the angle may change.