91Ó°ÊÓ

Find all values of \(z\) for which (a) \(e^{3 z}=1\) (b) \(e^{z^{2}}=1\) (c) \(e^{e^{z}}=1\).

Find the image of the region \(0 \leq x \leq \pi, y \geq 0\), for the transformation (a) \(w=e^{i z}\) (b) \(w=i e^{i z}\) (c) \(w=i e^{-i z}\).

(a) Show that both \(\sin z\) and \(\cos z\) are unbounded on the ray \(\operatorname{Arg} z=\theta\), \(0<|\theta|<\pi\) (b) Show that \(\sin z\) is bounded only on sets contained in a horizontal strip.

Show that the image of the disk \(|z| \leq 1\) under the transformation \(w=e^{z}\) is contained in the annulus \(1 / e \leq|w| \leq e\).

Find the image of straight lines parallel to the coordinate axes for the function (a) \(w=\log (i z)\) (b) \(w=\log (-i z)+1\) (c) \(w=\log z^{2}\)

(a) Separate \(e^{1 / z}, z \neq 0\) into its real and imaginary parts. (b) Show that \(\left|e^{1 / z}\right|\) is bounded in the region \(|z| \geq \epsilon, \epsilon>0\)

(a) Prove that \(e^{i z}\) is periodic, with period \(2 \pi\). (b) For an arbitrary nonzero complex number \(a\), show that \(e^{a z}\) is periodic, and find its period.

Show that (a) \(|\sin z|^{2}=\sin ^{2} x+\sinh ^{2} y\) (b) \(|\cos z|^{2}=\cos ^{2} x+\sinh ^{2} y\).

Find all the points of discontinuity of (i) \(f(z)=\log \left(z^{2}-1\right)\) (ii) \(f(z)=\operatorname{Arg}\left(z^{2}\right)\) (iii) \(f(z)=\log \left(z^{3}-1\right)\) (iv) \(f(z)=\operatorname{Arg}\left(z^{3}\right)\) (v) \(f(z)=\sqrt{z^{2}+1}\) (vi) \(f(z)=\sqrt{z^{2}-1}\).

$$ \text { Prove that } \tanh z=(\sinh z) /(\cosh z) \text { is periodic, with period } \pi i \text { . } $$