Practice Questions for CSIR NET Complex Analysis : Unit disk Automorphism, Schwarz Lemma II

11. Define

\( \begin{aligned} & H^{+}=\{z \in \mathbb{C}: y>0\} \\ & H^{-}=\{z \in \mathbb{C}: y<0\} \\ & L^{+}=\{z \in \mathbb{C}: x>0\} \\ & L^{-}=\{z \in \mathbb{C}: x<0\} \end{aligned} \)

The function \(f(z)=\frac{z}{3 z+1}\)

(a.) Maps \(H^{+}\) onto \(H^{+}\) and \(H^{-}\) onto \(H^{-}\)
(b.) Maps \(H^{+}\) onto \(H^{-}\) and \(H^{-}\) onto \(H^{+}\)
(c.) Maps \(H^{+}\) onto \(L^{+}\) and \(H^{-}\) onto \(L^{-}\)
(d.) Maps \(H^{+}\) onto \(L^{-}\) and \(H^{-}\) onto \(L^{+}\)

12. The region \(\{z \in \mathbb{C}:-1<\operatorname{Re}(z)<1\}\) can be mapped conformally onto

(a.) the upper half plane.
(b.) The annulus
(c.) the plane with two pts removed.
(d.) the punctured disk.

13. The transformation \(w=\frac{z-i}{z+i}\) maps

(a.) the upper half plane to itself
(b.) the upper half plane to lower half plane.
(c.) the upper half plane to unit disk
(d.) the unit disk to itself

14. An example of map which is conformal in the whole of its domain of definition is

(a.) \(f(z)=z^{2}\)
(b.) \(f(z)=e^{z+2}\)
(c.) \(f(z)=\cos z\)
(d.) \(f(z)=e^{z^{2}}\)

15. Let \(f: \mathbb{C} \backslash\{3 i\} \rightarrow \mathbb{C}\) be defined by \(f(z)=\frac{z-i}{i z+3}\). Which of the following statements about \(f\) is FALSE?

(a.) \(f\) is conformal on \(\mathbb{C} \backslash\{3 i\}\)
(b.) \(f\) maps circles in \(\mathbb{C} \backslash\{3 i\}\) onto circles in \(\mathbb{C}\)
(c.) All the fixed points of \(f\) are in the region \(\{z \in \mathbb{C}: \operatorname{Im}(z)>0\}\)
(d.) There is no straight line \(\mathbb{C} \backslash\{3 i\}\) which is mapped onto a straight line in \(\mathbb{C}\) by \(f\)

16. The image of the region \(\{z \in \mathbb{C}: \operatorname{Re}(z)>\operatorname{Im}(z)>0\}\) under the mapping \(z \mapsto e^{z^{2}}\) is

(a.) \(\{w \in \mathbb{C}: \operatorname{Re}(w)>0, \operatorname{Im}(w)>0\}\)
(b.) \(\{w \in \mathbb{C}: \operatorname{Re}(w)>0, \operatorname{Im}(w)>0,|w|>1\}\)
(c.) \(\{w \in \mathbb{C}:|w|>1\}\)
(d.) \(\{w \in \mathbb{C}: \operatorname{Im}(w)>0,|w|>1\}\)

17. Consider the function \(f(z)=\frac{z^{2}+\alpha z}{(z+1)^{2}}\) and \(g(z)=\sinh \left(z-\frac{\pi}{2 \alpha}\right), \alpha \neq 0\).

(i) The residue of \(f(z)\) at its pole is equal to 1. Then the value of \(\alpha\) is

(a.) -1
(b.) 1
(c.) 2
(d.) 3

(ii) For the value of \(\alpha\) obtained in the function \(g(z)\) is not conformal at a point

(a.) \(\frac{\pi(1+3 i)}{6}\)
(b.) \(\frac{\pi(3+i)}{6}\)
(c.) \(\frac{2 \pi}{3}\)
(d.) \(\frac{i \pi}{2}\)

18. Under the transformation \(w=\sqrt{\frac{1-i z}{z-i}}\), the region \(D=\{z \in \mathbb{C}:|z|<1\}\) is transformed to

(a.) \(\{z \in \mathbb{C}: 0<\arg z<\pi\}\)
(b.) \(\{z \in \mathbb{C}:-\pi<\arg z<0\}\)
(c.) \(\{z \in \mathbb{C}: 0<\arg z<\frac{\pi}{2}\) or \(\pi<\arg z<\frac{3 \pi}{2}\}\)
(d.) \(\{z \in \mathbb{C}: \frac{\pi}{2}<\arg z<\pi\) or \(\frac{3 \pi}{2}<\arg z<2 \pi\}\)

19. Let \(f\) be a bilinear transformation that maps 1 to 1, \(i\) to \(\infty\), and \(-i\) to 0 then \(f(1)\) is equal to

(a.) -2
(b.) -1
(c.) \(i\)
(d.) \(-i\)

20. Let \(G_{1}\) and \(G_{2}\) be the images of the disc \(\{z \in \mathbb{C}:|z+1|<1\}\) under the transformations \(w=\frac{(1-i) z+2}{(1+i) z+2}\) and \(w=\frac{(1+i) z+2}{(1-i) z+2}\) resp. then.

(a.) \(G_{1}=\{w \in \mathbb{C}: \operatorname{Im}(w)<0\}\) and \(G_{2}=\{w \in \mathbb{C}: \operatorname{Im}(w)>0\}\)
(b.) \(G_{1}=\{w \in \mathbb{C}: \operatorname{Im}(w)>0\}\) and \(G_{2}=\{w \in \mathbb{C}: \operatorname{Im}(w)<0\}\)
(c.) \(G_{1}=\{w \in \mathbb{C}:|w|>2\}\) and \(G_{2}=\{w \in \mathbb{C}:|w|<2\}\)
(d.) \(G_{1}=\{w \in \mathbb{C}:|w|<2\}\) and \(G_{2}=\{w \in \mathbb{C}:|w|>2\}\)