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> Show that $f:S=\{z\in\mathbb C:Re(z)\in[-\pi,\pi]\}\to\mathbb C:z\to e^{z}$ is bounded.
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**My Attempt:** $|f(x+iy)|=|e^x(\cos y+i\sin y)|=e^x$ is bounded on $\forall~ x\in[-\pi,\pi]\implies f$ is bounded on $S.$

Am I right?