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 $$\log z$$.

Definition. If any value of $$\log z$$ is equal to $$\zeta$$, we call $$z$$ the exponential of $$\zeta$$ and write $z = \exp \zeta.$

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.

Theorem. The exponential function $$\exp \zeta$$ is a one-valued function of $$\zeta$$.

For suppose that $z_{1} = r_{1}(\cos\theta_{1} + i\sin\theta_{1}),\quad z_{2} = r_{2}(\cos\theta_{2} + i\sin\theta_{2})$ 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 $\log r_{1} = \log r_{2},\quad \theta_{1} + 2m\pi = \theta_{2} + 2n\pi.$ Thus $$r_1 = r_2$$, and $$\theta_{1}$$ and $$\theta_{2}$$ differ by a multiple of $$2\pi$$. Hence $$z_{1} = z_{2}$$.

Corollary. If $$\zeta$$ is real then $$\exp \zeta = e^{\zeta}$$, the real exponential function of $$\zeta$$ defined in Ch. IX.

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 $r = e^{\xi},\quad \theta = \eta – 2m\pi;$ 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{aligned} \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{aligned} 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 $$\zeta_{2}$$.