complex z, lzl<1, map z to e^z, how many connected parts

Forum for the GRE subject test in mathematics.
cathy_liping
Posts: 9
Joined: Sat Oct 01, 2011 11:21 pm

complex z, lzl<1, map z to e^z, how many connected parts

Postby cathy_liping » Wed Oct 26, 2011 3:57 am

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!

owlpride
Posts: 204
Joined: Fri Jan 29, 2010 2:01 am

Re: complex z, lzl<1, map z to e^z, how many connected parts

Postby owlpride » Wed Oct 26, 2011 7:51 am

One?

The exponential function is continuous, the open disk |z|<1 is connected, and the continuous image of a connected set is connected.

Topoltergeist
Posts: 44
Joined: Tue Aug 09, 2011 6:18 pm

Re: complex z, lzl<1, map z to e^z, how many connected parts

Postby Topoltergeist » Wed Oct 26, 2011 5:41 pm

Here are my musings on the subject. I hope you find it useful.

The exponential map on complex numbers is similar to the Cartesian-polar 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.)

cathy_liping
Posts: 9
Joined: Sat Oct 01, 2011 11:21 pm

Re: complex z, lzl<1, map z to e^z, how many connected parts

Postby cathy_liping » Tue Nov 01, 2011 7:00 am

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 Cartesian-polar 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.)


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?




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