An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 82. The logarithm. 137
returned to the same leaf. In fig. 6, (p decreases from the value qpO to a
determinate negative value, then increases passing through zero up to a
value greater than zr and then decreases to the value (p0. Something
similar to this will occur along every other such curve even when
there are different pairs of intersections with the positive axis of
abscissae.
If on the other hand we make the argument S describe a finite
closed curve, not crossing itself but including the origin - = 0 and
so meeting the positive axis of abscissae in an odd number of points;
the value of z(z), as r and
Yi d g?(p vary continuously, will on
the return to the point A
come to be different from the
initial one by 2izX. In the
adjoining figure let z travel
_____/ __, f x from the point A so as to
\ ~o f ^^~ / \have the enclosed space on
the left, then (p increases from
>(o up to 2 z, becomes then
Fig. 7. > 2r, decreases to a value
between 2z and -3-z and again
increases, ultimately passing beyond 2z to the value cp +- 2. Thus
the value of the logarithm is Z(r) + i((p0 + 2z), while it was initially
l(r) + i p0.
We can extend these considerations to curves that repeatedly
go round the origin, crossing
]7 themselves in doing so, and
/-~/ ----~ _ ~formulate the following rule:
If in a determinately directed
circuit the positive axis of abscissae is crossed n times from
^_[aA below upwards, the value of
1(z) is increased by 2izn, for,
each crossing shows that a
circuit is completed. The path
7^ig. a. between two crossings only
Fig. 8.
signifies that there has been
no circuit, when, between the two, the amplitude (p had a retrograde
motion, so that at an even or an odd number of points the curve
has cut the positive axis of abscissa from above downwards. If there
be mn such points, 2izm is to be deducted; therefore the value
of the logarithm changes by 2izr(n - m), when the numbers
of the crossings are respectively n and in.