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(Solved): 27. Let \( z=x+i y \) and \( w=u+i v \), and consider the mapping by the function \( w=f(z)= \) \( ...
27. Let \( z=x+i y \) and \( w=u+i v \), and consider the mapping by the function \( w=f(z)= \) \( \sinh z \). (i) Show that \( f \) maps the half-infinite strip \( x>0,-\pi / 20 \). (ii) Use the formula \( e^{\alpha+i \beta}=e^{\alpha}(\cos \beta+i \sin \beta) \) to derive the formula \( \sinh (x+i y)=\cos y \sinh x+ \) \( i \sin y \cosh x \). (iii) Use (ii) to show that \( f \) maps the vertical line segments \( x=\xi,-\pi / 2