« Section 7.1
Section 7.3 »

Section 7.2

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1. (a) \displaystyle{\lim_{n \to \infty}z_n = -1 + \frac{1}{2}i}

(c) \displaystyle{\lim_{n \to \infty}z_n = 3}

2. (a) \displaystyle{\lim_{z \to i}(4z^3 - 6z + 3) = 3 - 10i}

(c) \displaystyle{\lim_{w \to 3i}\frac{w^2 + 9}{w - 3i} = 6i}

3. (a) f'(z) = 6z - 30z^4

(c) \displaystyle{f'(z) = -2z(z - 4i)e^{-z^2} + e^{-z^2}}

5. (a) The result follows from

\displaystyle{e^z = e^{x + yi} = e^xe^{yi} = e^x(\cos(y) + i\sin(y)).}

(b) \left|e^z\right| = e^x , \mathrm{arg}\left(e^z\right) = y

11. (a) For all z ,

\displaystyle{\sin(z) = \sum_{n=0}^\infty \frac{(-1)^nz^{2n+1}}{(2n+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \frac{z^9}{9!} - \cdots .}

(b) For all z ,

\displaystyle{\cos(z) = \sum_{n=0}^\infty \frac{(-1)^nz^{2n}}{2n!} = 1 - \frac{z^2}{2} + \frac{z^4}{4!} - \frac{z^6}{6!} + \frac{z^8}{8!} - \cdots .}

15. (b) Hint: Consider what happens if z = x + yi where y is not between -\pi and \pi .

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