85. Limits of complex functions and series of complex terms.
In this chapter we have, up to the present, concerned ourselves only with real functions of \(n\) and series all of whose terms are real. There is however no difficulty in extending our ideas and definitions to the case in which the functions or the terms of the series are complex.
Suppose that \(\phi(n)\) is complex and equal to \[\rho(n) + i\sigma(n),\] where \(\rho(n)\), \(\sigma(n)\) are real functions of \(n\). Then if \(\rho(n)\) and \(\sigma(n)\) converge respectively to limits \(r\) and \(s\) as \(n \to \infty\), we shall say that \(\phi(n)\) converges to the limit \(l = r + is\), and write \[\lim\phi(n) = l.\] Similarly, when \(u_{n}\) is complex and equal to \(v_{n} + iw_{n}\), we shall say that the series \[u_{1} + u_{2} + u_{3} + \dots\] is convergent and has the sum \(l = r + is\), if the series \[v_{1} + v_{2} + v_{3} + \dots,\quad w_{1} + w_{2} + w_{3} + \dots\] are convergent and have the sums \(r\), \(s\) respectively.
To say that \(u_{1} + u_{2} + u_{3} + \dots\) is convergent and has the sum \(l\) is of course the same as to say that the sum \[s_{n} = u_{1} + u_{2} + \dots + u_{n} = (v_{1} + v_{2} + \dots + v_{n}) + i(w_{1} + w_{2} + \dots + w_{n})\] converges to the limit \(l\) as \(n \to \infty\).
In the case of real functions and series we also gave definitions of divergence and oscillation, finite or infinite. But in the case of complex functions and series, where we have to consider the behaviour both of \(\rho(n)\) and of \(\sigma(n)\), there are so many possibilities that this is hardly worth while. When it is necessary to make further distinctions of this kind, we shall make them by stating the way in which the real or imaginary parts behave when taken separately.
86.
The reader will find no difficulty in proving such theorems as the following, which are obvious extensions of theorems already proved for real functions and series.
(1) If \(\lim\phi(n) = l\) then \(\lim\phi(n + p) = l\) for any fixed value of \(p\).
(2) If \(u_{1} + u_{2} + \dots\) is convergent and has the sum \(l\), then \(a + b + c + \dots + k + u_{1} + u_{2} + \dots\) is convergent and has the sum \(a + b + c + \dots + k + l\), and \(u_{p+1} + u_{p+2} + \dots\) is convergent and has the sum \(l – u_{1} – u_{2} – \dots – u_{p}\).
(3) If \(\lim\phi(n) = l\) and \(\lim\psi(n) = m\), then \[\lim\{\phi(n) + \psi(n)\} = l + m.\]
(4) If \(\lim\phi(n) = l\), then \(\lim k\phi(n) = kl\).
(5) If \(\lim\phi(n) = l\) and \(\lim\psi(n) = m\), then \(\lim \phi(n)\psi(n) = lm\).
(6) If \(u_{1} + u_{2} + \dots\) converges to the sum \(l\), and \(v_{1} + v_{2} + \dots\) to the sum \(m\), then \((u_{1} + v_{1}) + (u_{2}+ v_{2}) + \dots\) converges to the sum \(l + m\).
(7) If \(u_{1} + u_{2} + \dots\) converges to the sum \(l\) then \(ku_{1} + ku_{2} + \dots\) converges to the sum \(kl\).
(8) If \(u_{1} + u_{2} + u_{3} + \dots\) is convergent then \(\lim u_{n} = 0\).
(9) If \(u_{1} + u_{2} + u_{3} + \dots\) is convergent, then so is any series formed by grouping the terms in brackets, and the sums of the two series are the same.
As an example, let us prove theorem (5). Let \[\phi(n) = \rho(n) + i\sigma(n),\quad \psi(n) = \rho'(n) + i\sigma'(n),\quad l = r + is,\quad m = r’ + is’.\]
Then \[\rho(n) \to r,\quad \sigma(n) \to s,\quad \rho'(n) \to r’,\quad \sigma'(n) \to s’.\]
But \[\phi(n)\psi(n) = \rho\rho’ – \sigma\sigma’ + i(\rho\sigma’ + \rho’\sigma),\] and \[\rho\rho’ – \sigma\sigma’ \to rr’ – ss’,\quad \rho\sigma’ + \rho’\sigma \to rs’ + r’s;\] so that \[\phi(n)\psi(n) \to rr’ – ss’ + i(rs’ + r’s),\] i.e. \[\phi(n)\psi(n) \to (r + is)(r’ + is’) = lm.\]
The following theorems are of a somewhat different character.
(10) In order that \(\phi(n) = \rho(n) + i\sigma(n)\) should converge to zero as \(n \to \infty\), it is necessary and sufficient that \[|\phi(n)| = \sqrt{\{\rho(n)\}^{2} + \{\sigma(n)\}^{2}}\] should converge to zero.
If \(\rho(n)\) and \(\sigma(n)\) both converge to zero then it is plain that \(\sqrt{\rho^{2} + \sigma^{2}}\) does so. The converse follows from the fact that the numerical value of \(\rho\) or \(\sigma\) cannot be greater than \(\sqrt{\rho^{2} + \sigma^{2}}\).
(11) More generally, in order that \(\phi(n)\) should converge to a limit \(l\), it is necessary and sufficient that \[|\phi(n) – l|\] should converge to zero.
For \(\phi(n) – l\) converges to zero, and we can apply (10).
(12) Theorems 1 and 2 of §§ 83-84 are still true when \(\phi(n)\) and \(u_{n}\) are complex.
We have to show that the necessary and sufficient condition that \(\phi(n)\) should tend to \(l\) is that \[\begin{equation*} |\phi(n_{2}) – \phi(n_{1})| < \epsilon \tag{1} \end{equation*}\] when \(n_{2} > n_{1} \geq n_{0}\).
If \(\phi(n) \to l\) then \(\rho(n) \to r\) and \(\sigma(n) \to s\), and so we can find numbers \(n_{0}’\) and \(n_{0}”\) depending on \(\epsilon\) and such that \[|\rho(n_{2}) – \rho(n_{1})| < \tfrac{1}{2}\epsilon,\quad |\sigma(n_{2}) – \sigma(n_{1})| < \tfrac{1}{2}\epsilon,\] the first inequality holding when \(n_{2} > n_{1} \geq n_{0}’\), and the second when \(n_{2} > n_{1} \geq n_{0}”\). Hence \[|\phi(n_{2}) – \phi(n_{1})| \leq |\rho(n_{2}) – \rho(n_{1})| + |\sigma(n_{2}) – \sigma(n_{1})| < \epsilon\] when \(n_{2} > n_{1} \geq n_{0}\), where \(n_{0}\) is the greater of \(n_{0}’\) and \(n_{0}”\). Thus the condition (1) is necessary. To prove that it is sufficient we have only to observe that \[|\rho(n_{2}) – \rho(n_{1})| \leq |\phi(n_{2}) – \phi(n_{1})| < \epsilon\] when \(n_{2} > n_{1} \geq n_{0}\). Thus \(\rho(n)\) tends to a limit \(r\), and in the same way it may be shown that \(\sigma(n)\) tends to a limit \(s\).
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