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Homework 5 (MATH 1199 Fall 2019)
1. Compute...
(a) $2^{1-i}$
(b) $(P.V.) (1+i)^{2i}$
(c) $1^i$
(d) $(P.V.) (-2)^{-i}$
(e) $\arcsin(-i)$
(f) $\mathrm{Arccos} \left( \dfrac{1}{2} \right)$
2. Use the definitions of $\sin$, $\cos$, $\sinh$, and $\cosh$ to show...
(a) $\dfrac{\mathrm{d}}{\mathrm{d}z} \cos(z) = -\sin(z)$
(b) (double angle identity) $\sin(2z)=2\sin(z)\cos(z)$
(c) $\dfrac{\mathrm{d}}{\mathrm{d}z} \sinh(z)=\cosh(z)$
(d) $\cosh(iz)=\cos(z)$
3. Define $\tan(z) = \dfrac{\sin(z)}{\cos(z)}$.
(a) Show that $\tan(z) = \dfrac{1}{i} \left( \dfrac{e^{iz}-e^{-iz}}{e^{iz}+e^{-iz}} \right)$.
(b) Resembling p.39-40 in the notes, let $z=\tan(w)$ so that $w=\arctan(z)$. Express $\arctan(z)$ in terms of the logarithm. Hint: it should again become a quadratic in the variable $v=e^{iw}$ (hence: $\frac{1}{v}=e^{-iw}$), but it should not require quadratic formula to solve.
