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Homework 4 (MATH 1199 Fall 2019)
1. Write an $\epsilon$-$\delta$ proof that the following functions are continuous.
(a) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \rightarrow \mathbb{C} \\
f(z) = 3z-7
\end{array} \right.$
(b) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \rightarrow \mathbb{C} \\
f(z) = 7
\end{array} \right.$
(c) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \rightarrow \mathbb{C} \\
f(z)=\mathrm{Re}(z)
\end{array} \right.$
2. Use the limit definition of the derivative to compute
(a) $\dfrac{\mathrm{d}}{\mathrm{d}z} 2z^2+z$
(b) $\dfrac{\mathrm{d}}{\mathrm{d}z} z^3-5z$
3. Verify that the Cauchy-Riemann equations hold for the given function.
(a) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \rightarrow \mathbb{C} \\
f(z) = 3z^2+1 \end{array} \right.$
(b) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \rightarrow \mathbb{C} \\
f(z) = z^5-13z^4+3z^2-10
\end{array} \right.$
(c) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \setminus \{0\} \rightarrow \mathbb{C} \\
f(z) = \dfrac{1}{z}
\end{array} \right.$
(d) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \rightarrow \mathbb{C} \\
f(x+iy) = e^x e^{iy}, \quad x, y \in \mathbb{R} \end{array} \right.$
4. Explain why the function is not differentiable everywhere using an appropriate calculation (i.e. the Corollary on p.31 of the notes).
(a) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \rightarrow \mathbb{C} \\
f(z) = z + \overline{z}
\end{array} \right.$
(b) $\left\{ \begin{array}{ll}
f \colon \mathbb{C} \rightarrow \mathbb{C} \\
f(x+iy) = e^x e^{-iy}, \quad x, y \in \mathbb{R}
\end{array} \right.$
5. Find...
(a) $\exp \left( -7 + 9\pi i \right)$
(b) $\exp \left( \dfrac{-17+\pi i}{4} \right)$
(c) $\mathrm{Log}(-ei)$
(d) $\mathrm{Log}(1+i)$
(e) $\mathrm{log}(\sqrt{3}-i)$ (note: lowercase $\mathrm{log}$ here!! "multivalued"!!!)
(f) $\mathrm{log}(1-i)$
