222–224. The exponential function

222. The exponential function.

In Ch. IX we defined a function $e^y$ of the real variable $y$ as the inverse of the function $y=\log x$. It is naturally suggested that we should define a function of the complex variable $z$ which is the inverse of the function $\log z$.

Thus $z=\exp \zeta$ if $\zeta =\log z$. It is certain that to any given value of $z$ correspond infinitely many different values of $\zeta$. It would not be unnatural to suppose that, conversely, to any given value of $\zeta$ correspond infinitely many values of $z$, or in other words that $\exp \zeta$ is an infinitely many-valued function of $\zeta$. This is however not the case, as is proved by the following theorem.

For suppose that $$z_1=r_1(\cos {\theta }_1+i\sin {\theta }_1),z_2=r_2(\cos {\theta }_2+i\sin {\theta }_2)$$ are both values of $\exp \zeta$. Then $$\zeta =\log z_1=\log z_2,$$ and so $$\log r_1+i({\theta }_1+2m\pi )=\log r_2+i({\theta }_2+2n\pi ),$$ where $m$ and $n$ are integers. This involves $$\log r_1=\log r_2,{\theta }_1+2m\pi ={\theta }_2+2n\pi .$$ Thus $r_1=r_2$, and ${\theta }_1$ and ${\theta }_2$ differ by a multiple of $2\pi$. Hence $z_1=z_2$.

For if $z=e^{\zeta }$ then $\log z=\zeta$, one of the values of $\log z$ is $\zeta$. Hence $z=\exp \zeta$.

 

223. The value of $\exp \zeta$.

Let $\zeta =\xi +i\eta$ and $$z=\exp \zeta =r(\cos \theta +i\sin \theta ).$$ Then $$\xi +i\eta =\log z=\log r+i(\theta +2m\pi ),$$ where $m$ is an integer. Hence $\xi =\log r$, $\eta =\theta +2m\pi$, or $$r=e^{\xi },\theta =\eta –2m\pi ;$$ and accordingly $$\exp (\xi +i\eta )=e^{\xi }(\cos \eta +i\sin \eta ).$$

If $\eta =0$ then $\exp \xi =e^{\xi }$, as we have already inferred in § 222. It is clear that both the real and the imaginary parts of $\exp (\xi +i\eta )$ are continuous functions of $\xi$ and $\eta$ for all values of $\xi$ and $\eta$.

Let ${\zeta }_1={\xi }_1+i{\eta }_1$, ${\zeta }_2={\xi }_2+i{\eta }_2$. Then $$\begin{array}{rl}\exp {\zeta }_1\times \exp {\zeta }_2& =e^{{\xi }_1}(\cos {\eta }_1+i\sin {\eta }_1)\times e^{{\xi }_2}(\cos {\eta }_2+i\sin {\eta }_2)\\ & =e^{{\xi }_1+{\xi }_2}\{\cos ({\eta }_1+{\eta }_2)+i\sin ({\eta }_1+{\eta }_2)\}\\ & =\exp ({\zeta }_1+{\zeta }_2).\end{array}$$ The exponential function therefore satisfies the functional relation $f({\zeta }_1+{\zeta }_2)=f({\zeta }_1)f({\zeta }_2)$, an equation which we have proved already (§ 205) to be true for real values of ${\zeta }_1$ and ${\zeta }_2$.

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$\leftarrow$ 221. The values of the logarithmic function

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225–226. The general power $a^z$ $\to$