Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Example of a bijective conformal map between $$S_{1} = \{re^{i\theta} \ : \ 0 < r < 2 \ , 0 < \theta <\pi/4\}$$ and $$S_{2} = \{ x +iy : 0 < y <1\}$$

share|improve this question
4  
What is the question? –  draks ... Sep 27 '12 at 8:40
add comment

closed as not a real question by Austin Mohr, Gigili, Belgi, Chris Eagle, Sasha Sep 28 '12 at 4:18

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

1 Answer

I assume that the question is to find a bijective conformal map, or a biholomorphic map, between the "pizza-slice" $S_1$ and the infinite strip $S_2$. You can do that basically in three steps.

(1) The complex logarithm $$f(z)=\log(z)=\log|z|+{\rm i}\arg(z)$$ will map $S_1$ to some semi-infinite strip.

(2) The hyperbolic cosine $$g(z)=\cosh(z)={1\over 2}({\rm e}^z+{\rm e}^{-z})=\cosh(x)\cos(y)+{\rm i}\sinh(x)\sin(y)$$ will map the semi-infinite strip $\{(x,y):x>0, 0<y<\pi\}$ to the upper half plane.

(3) The complex logarithm (again!) will map the upper halfplane to some infinite strip.

Now, you need to identify the semi-infinite strip in (1) and the infinite strip in (3). You also need to add some linear maps to bridge the steps, since the semi-infinite strip you get in (1) is not the one you start with in (2), and the infinite strip you get in (3) is not the same as $S_2$. Details are left to the interested reader.

share|improve this answer
add comment