i know complex z is a unit circle, and when z is real axis, this transformation will map z to e^x, i think this is one connected part.
but what about if z is imaginary axis, what is the transformation through e^z
thank you in advance!
complex z, lzl<1, map z to e^z, how many connected parts

 Posts: 9
 Joined: Sat Oct 01, 2011 11:21 pm
Re: complex z, lzl<1, map z to e^z, how many connected parts
One?
The exponential function is continuous, the open disk z<1 is connected, and the continuous image of a connected set is connected.
The exponential function is continuous, the open disk z<1 is connected, and the continuous image of a connected set is connected.

 Posts: 44
 Joined: Tue Aug 09, 2011 6:18 pm
Re: complex z, lzl<1, map z to e^z, how many connected parts
Here are my musings on the subject. I hope you find it useful.
The exponential map on complex numbers is similar to the Cartesianpolar change of coordinates in 2D: $$e^{x+iy} = e^x(\cos y + i \sin y).$$ Horizontal lines such as $$z(x) = x + bi$$ will be mapped to rays extending from the origin, with the complex component $$b$$ determining the angle of inclination. Vertical lines such as $$z(y) = a + i y$$ will be mapped to circles centered at the origin. The real component, $$a$$ will determine the radius of the circle. The imaginary axis gets mapped to the unit circle. (In fact, the exponential map is a covering map of the circle from a universal cover.)
The exponential map on complex numbers is similar to the Cartesianpolar change of coordinates in 2D: $$e^{x+iy} = e^x(\cos y + i \sin y).$$ Horizontal lines such as $$z(x) = x + bi$$ will be mapped to rays extending from the origin, with the complex component $$b$$ determining the angle of inclination. Vertical lines such as $$z(y) = a + i y$$ will be mapped to circles centered at the origin. The real component, $$a$$ will determine the radius of the circle. The imaginary axis gets mapped to the unit circle. (In fact, the exponential map is a covering map of the circle from a universal cover.)

 Posts: 9
 Joined: Sat Oct 01, 2011 11:21 pm
Re: complex z, lzl<1, map z to e^z, how many connected parts
thank you for your answer first! i think the real axis and imaginary axis are two connected components. and what about other parts except the real axis and imaginary axis?Topoltergeist wrote:Here are my musings on the subject. I hope you find it useful.
The exponential map on complex numbers is similar to the Cartesianpolar change of coordinates in 2D: $$e^{x+iy} = e^x(\cos y + i \sin y).$$ Horizontal lines such as $$z(x) = x + bi$$ will be mapped to rays extending from the origin, with the complex component $$b$$ determining the angle of inclination. Vertical lines such as $$z(y) = a + i y$$ will be mapped to circles centered at the origin. The real component, $$a$$ will determine the radius of the circle. The imaginary axis gets mapped to the unit circle. (In fact, the exponential map is a covering map of the circle from a universal cover.)