Exponential function

Definition 2..1   $w = e^{z} = e^{x}(\cos{y} + i\sin{y})$ is called an exponential function and expressed as $w = e^{z}$ or $w = \exp{z}$.

Theorem 2..1 (1)   $e^{z_{1} + z_{2}} = e^{z_{1}}e^{z_{2}}, (e^{z})^{n} = e^{nz} (n is integer)$
(2) $e^{z}$ has a period of $2\pi i$.

Proof (1) Let $z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}$. Then

$$e^{z_{1} + z_{2}} = e^{x_{1} + x_{2} + i(y_{1} + y_{2})}$$

$$= e^{x_{1} + x_{2}}(\cos{(y_{1} + y_{2})} + i\sin(y_{1} + y_{2}))$$

$$= e^{x_{1} + x_{2}}[\cos{y_{1}}\cos{y_{2}} - \sin{y_{1}}\sin{y_{2}} + i(\sin{y_{1}}\cos{y_{2}} + \cos{y_{1}}\sin{y_{2}})]$$

$$= e^{x_{1} + x_{2}}[(\cos{y_{1}} + i\sin{y_{1}})(\cos{y_{2}}+i\sin{y_{2}})]$$

$$= e^{x_{1}}(\cos{y_{1}} + i\sin{y_{1}})e^{x_{2}}(\cos{y_{2}}+i\sin{y_{2}})$$

$$= e^{z_{1}}e^{z_{2}}$$

(2) When $e^{z + c} = e^{z}$ holds, the function $e^{z}$ is said to have a period $c$. Then we let $e^{z + c} = e^{z}$. Since $e^{z}e^{c} = e^{z}$, we have $e^{c} = 1$. Let $c = a + bi$. Then $e^{c} = e^{a + bi} = e^{a}(\cos{b} + i\sin{b}) = 1$ implies $e^{a}\cos{b} = 1, e^{a}\sin{b} = 0$. Thus, $a = 0, \cos{b} = 1, \sin{b} = 0$. $b$ that satisfies this, $b = 2n\pi ( n = 0, \pm1,\pm2,\ldots)$. Therefore, $e^{z}$ has a period $2\pi i$.

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 $\tan{z}$.