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 \(\zeta_{2}\).
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