Let us suppose that the polar coordinates of the point $$z = \zeta$$ are $$\rho$$ and $$\phi$$, so that $\zeta = \rho(\cos\phi + i\sin\phi).$ We suppose for the present that $$-\pi < \phi < \pi$$, while $$\rho$$ may have any positive value. Thus $$\zeta$$ may have any value other than zero or a real negative value.

The coordinates $$(x, y)$$ of any point on the path $$C$$ are functions of $$t$$, and so also are its polar coordinates $$(r, \theta)$$. Also \begin{aligned} \log \zeta &= \int_{C} \frac{dz}{z} = \int_{C} \frac{dx + i\, dy}{x + iy} \\ &= \int_{t_{0}}^{t_{1}} \frac{1}{x + iy} \left(\frac{dx}{dt} + i\frac{dy}{dt}\right) dt,\end{aligned} in virtue of the definitions of § 219. But $$x = r\cos\theta$$, $$y = r\sin\theta$$, and \begin{aligned} \frac{dx}{dt} + i\frac{dy}{dt} &= \left(\cos\theta\, \frac{dr}{dt} – r\sin\theta\, \frac{d\theta}{dt}\right) + i\left(\sin\theta\, \frac{dr}{dt} + r\cos\theta\, \frac{d\theta}{dt}\right) \\ &= (\cos\theta + i\sin\theta) \left(\frac{dr}{dt} + ir\frac{d\theta}{dt}\right);\end{aligned} so that $\log \zeta = \int_{t_{0}}^{t_{1}} \frac{1}{r}\, \frac{dr}{dt}\, dt + i\int_{t_{0}}^{t_{1}} \frac{d\theta}{dt}\, dt = [\log r] + i[\theta],$ where $$[\log r]$$ denotes the difference between the values of $$\log r$$ at the points corresponding to $$t = t_{1}$$ and $$t = t_{0}$$, and $$[\theta]$$ has a similar meaning.

It is clear that $[\log r] = \log \rho – \log 1 = \log \rho;$ but the value of $$[\theta]$$ requires a little more consideration. Let us suppose first that the path of integration is the straight line from $$1$$ to $$\zeta$$. The initial value of $$\theta$$ is the amplitude of $$1$$, or rather one of the amplitudes of $$1$$, viz. $$2k\pi$$, where $$k$$ is any integer. Let us suppose that initially $$\theta = 2k\pi$$. It is evident from the figure that $$\theta$$ increases from $$2k\pi$$ to $$2k\pi + \phi$$ as $$t$$ moves along the line. Thus $[\theta] = (2k\pi + \phi) – 2k\pi = \phi,$ and, when the path of integration is a straight line, $$\log \zeta = \log \rho + i\phi$$.

We shall call this particular value of $$\log \zeta$$ the principal value. When $$\zeta$$ is real and positive, $$\zeta = \rho$$ and $$\phi = 0$$, so that the principal value of $$\log \zeta$$ is the ordinary logarithm $$\log \zeta$$. Hence it will be convenient in general to denote the principal value of $$\log \zeta$$ by $$\log \zeta$$. Thus $\log \zeta = \log \rho + i\phi,$ and the principal value is characterised by the fact that its imaginary part lies between $$-\pi$$ and $$\pi$$.

Next let us consider any path (such as those shown in Fig. 56) such that the area or areas included between the path and the straight line from $$1$$ to $$\zeta$$ does not include the origin.

It is easy to see that $$[\theta]$$ is still equal to $$\phi$$. Along the curve shown in the figure by a continuous line, for example, $$\theta$$, initially equal to $$2k\pi$$, first decreases to the value $2k\pi – XOP$ and then increases again, being equal to $$2k\pi$$ at $$Q$$, and finally to $$2k\pi + \phi$$. The dotted curve shows a similar but slightly more complicated case in which the straight line and the curve bound two areas, neither of which includes the origin. Thus if the path of integration is such that the closed curve formed by it and the line from $$1$$ to $$\zeta$$ does not include the origin, then $\log \zeta = \log \zeta = \log \rho + i\phi.$

On the other hand it is easy to construct paths of integration such that $$[\theta]$$ is not equal to $$\phi$$. Consider, for example, the curve indicated by a continuous line in Fig. 57. If $$\theta$$ is initially equal to $$2k\pi$$, it will have increased by $$2\pi$$ when we get to $$P$$ and by $$4\pi$$ when we get to $$Q$$; and its final value will be $$2k\pi + 4\pi + \phi$$, so that $$[\theta] = 4\pi + \phi$$ and $\log \zeta = \log \rho + i(4\pi + \phi).$

In this case the path of integration winds twice round the origin in the positive sense. If we had taken a path winding $$k$$ times round the origin we should have found, in a precisely similar way, that $$[\theta] = 2k\pi+ \phi$$ and $\log \zeta = \log \rho + i(2k\pi + \phi).$ Here $$k$$ is positive. By making the path wind round the origin in the opposite direction (as shown in the dotted path in Fig. 57), we obtain a similar series of values in which $$k$$ is negative. Since $$|\zeta | = \rho$$, and the different angles $$2k\pi + \phi$$ are the different values of $$\operatorname{am} \zeta$$, we conclude that every value of $$\log |\zeta| + i\operatorname{am} \zeta$$ is a value of $$\log \zeta$$; and it is clear from the preceding discussion that every value of $$\log \zeta$$ must be of this form.

We may summarise our conclusions as follows:

the general value of $$\log \zeta$$ is $\log |\zeta| + i\operatorname{am} \zeta = \log \rho + i(2k\pi + \phi),$ where $$k$$ is any positive or negative integer. The value of $$k$$ is determined by the path of integration chosen. If this path is a straight line then $$k = 0$$ and $\log \zeta = \log \zeta = \log \rho + i\phi.$

In what precedes we have used $$\zeta$$ to denote the argument of the function $$\log \zeta$$, and $$(\xi, \eta)$$ or $$(\rho, \phi)$$ to denote the coordinates of $$\zeta$$; and $$z$$, $$(x, y)$$, $$(r, \theta)$$ to denote an arbitrary point on the path of integration and its coordinates. There is however no reason now why we should not revert to the natural notation in which $$z$$ is used as the argument of the function $$\log z$$, and we shall do this in the following examples.

Example XCIII
1. We supposed above that $$-\pi < \theta < \pi$$, and so excluded the case in which $$z$$ is real and negative. In this case the straight line from $$1$$ to $$z$$ passes through $$0$$, and is therefore not admissible as a path of integration. Both $$\pi$$ and $$-\pi$$ are values of $$\operatorname{am} z$$, and $$\theta$$ is equal to one or other of them: also $$r = -z$$. The values of $$\log z$$ are still the values of $$\log |z| + i\operatorname{am} z$$, viz. $\log (-z) + (2k + 1)\pi i,$ where $$k$$ is an integer. The values $$\log (-z) + \pi i$$ and $$\log (-z) – \pi i$$ correspond to paths from $$1$$ to $$z$$ lying respectively entirely above and entirely below the real axis. Either of them may be taken as the principal value of $$\log z$$, as convenience dictates. We shall choose the value $$\log (-z) + \pi$$ i corresponding to the first path.

2. The real and imaginary parts of any value of $$\log z$$ are both continuous functions of $$x$$ and $$y$$, except for $$x = 0$$, $$y = 0$$.

3. The functional equation satisfied by $$\log z$$. The function $$\log z$$ satisfies the equation $\begin{equation*} \log z_{1} z_{2} = \log z_{1} + \log z_{2}, \tag{1} \end{equation*}$ in the sense that every value of either side of this equation is one of the values of the other side. This follows at once by putting $z_{1} = r_{1}(\cos\theta_{1} + i\sin\theta_{1}),\quad z_{2} = r_{2}(\cos\theta_{2} + i\sin\theta_{2}),$ and applying the formula above. It is however not true that $\begin{equation*} \log z_{1}z_{2} = \log z_{1} + \log z_{2} \tag{2} \end{equation*}$ in all circumstances. If, e.g., $z_{1} = z_{2} = \tfrac{1}{2}(-1 + i\sqrt{3}) = \cos \tfrac{2}{3}\pi + i \sin \tfrac{2}{3}\pi,$ then $$\log z_{1} = \log z_{2} = \frac{2}{3}\pi i$$, and $$\log z_{1} + \log z_{2} = \frac{4}{3}\pi i$$, which is one of the values of $$\log z_{1}z_{2}$$, but not the principal value. In fact $$\log z_{1}z_{2} = -\frac{2}{3}\pi i$$.

An equation such as (1), in which every value of either side is a value of the other, we shall call a complete equation, or an equation which is completely true.

4. The equation $$\log z^{m} = m\log z$$, where $$m$$ is an integer, is not completely true: every value of the right-hand side is a value of the left-hand side, but the converse is not true.

5. The equation $$\log (1/z) = -\log z$$ is completely true. It is also true that $$\log (1/z) = -\log z$$, except when $$z$$ is real and negative.

6. The equation $\log \left(\frac{z – a}{z – b}\right) = \log (z – a) – \log (z – b)$ is true if $$z$$ lies outside the region bounded by the line joining the points $$z = a$$, $$z = b$$, and lines through these points parallel to $$OX$$ and extending to infinity in the negative direction.

7. The equation $\log \left(\frac{a – z}{b – z}\right) = \log \left(1 – \frac{a}{z}\right) – \log \left(1 – \frac{b}{z}\right)$ is true if $$z$$ lies outside the triangle formed by the three points $$O$$$$a$$$$b$$.

8. Draw the graph of the function $$\operatorname{I}(\log x)$$ of the real variable $$x$$. [The graph consists of the positive halves of the lines $$y = 2k\pi$$ and the negative halves of the lines $$y = (2k + 1)\pi$$.]

9. The function $$f(x)$$ of the real variable $$x$$, defined by $\pi f(x) = p\pi + (q – p)\operatorname{I}(\log x),$ is equal to $$p$$ when $$x$$ is positive and to $$q$$ when $$x$$ is negative.

10. The function $$f(x)$$ defined by $\pi f(x) = p\pi + (q – p)\operatorname{I}\{\log(x – 1)\} + (r – q)\operatorname{imag}(\log x)$ is equal to $$p$$ when $$x > 1$$, to $$q$$ when $$0 < x < 1$$, and to $$r$$ when $$x < 0$$.

11. For what values of $$z$$ is (i) $$\log z$$ (ii) any value of $$\log z$$ (a) real or (b) purely imaginary?

12. If $$z = x + iy$$ then $$\log\log z = \log R + i(\Theta + 2k’\pi)$$, where $R^{2} = (\log r)^{2} + (\theta + 2k\pi)^{2}$ and $$\Theta$$ is the least positive angle determined by the equations $\cos\Theta : \sin\Theta : 1 :: \log r : \theta + 2k\pi: \sqrt{(\log r)^{2} + (\theta + 2k\pi)^{2}}.$ Plot roughly the doubly infinite set of values of $$\log\log(1 + i\sqrt{3})$$, indicating which of them are values of $$\log\log(1 + i \sqrt{3})$$ and which of $$\log\log(1 + i\sqrt{3})$$.