meromorphic function

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Exercise2.3

1. Find a complex number that satisfies the following equation.

(a)

$e^{z} = 1$

(b)

$e^{z} = i$

(c)

$e^{z} = -2$

2. Express the following value in the form of $u + iv$.

(a)

$\sin{2i}$

(b)

$\sin{\left(\frac{\pi}{2} + i\right)}$

(c)

$\cos{\left(\frac{\pi}{3} - i\right)}$

(d)

$\tan{\left(\frac{\pi}{6} + 2i\right)}$

(e)

$\sin(iy)$

(f)

$\cos(iy)$

3. Prove the following formulas.

(a)

$1 + \tan^{2}{z} = \frac{1}{\cos^{2}{z}}$

(b)

$\sin{(-z)} = -\sin{z}$

(c)

$\cos(-z) = \cos{z}$

4. Show that for $\sin{z}$.

(a)

$\sin{z}$ has a period $2\pi$

(5) Find the period of $\tan{z}$.