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.

~ 182.

Integration of a rational function.

335

abscissae, to avoid having the lower limit + 1 on the branching section,
for in that case we should have to distinguish between its two banks.
To proceed from 1 to z in the same leaf, means to take a path
that, while otherwise arbitrary, does not cross this section. To proceed
from 1 to z and at the same time pass from the first into the
(7 + I)th leaf, means to take an arbitrary path which crosses this
line (k + -k') times in the direction from right to left and 7' times in
the opposite direction; to pass from the firstsnto the - (k + l)th leaf,
means to cross the section (k' + k) times from left to right and
k' times from right to left.
Remaining in the first leaf, we can as a particular case integrate
from 1 to z = x - iy, provided S is not in the second quadrant
(x < 0, y > 0), by proceeding first from 1 parallel to ordinates to y
and then parallel to abscissa to the value x.
Putting:
Z X0 o+'yo
rdx z d d dx-idy_ ( - iy) d. (x-iy dy
Jf i + Jy ix- Y+ x2 +
then (as in ~ 175, 4.) the first integral on the right is zero along the
first part of the path of integration, along the second it is either:
xo
-1 d l ((xO2+ y02)- 1(1+ 2) +i {ta ll
J x2 2 ^^ ^-(- -]-i tan — I. - tan-ly,
X+ — O 2 2 So
1
or:
-X- y X2 +82 /( + - yo2) ~ i {tan-1? -tan- o - z,
according as xo is positive or negative; the circular function here
always signifies a value between - - r and + - z. For in the
second case, as x passes through zero, tan-' y~ is led over continuously
~x~~ ~x
into the value tan- I YO- Zr.
We find for the second integral along the path parallel to
ordinates:
0
l+y2 D' 2 (1+ o) + itany-,.
Here the circular function signifies always a value between - -t and
+- ~e; therefore in the first leaf, (excluding Yo > 0, xo < 0), we have:
z=Xoq-iyo
jdz 1 l(x,,2 + y02) + itanll Y + 0 or, -- i
according as xt > O, or, < i.
according as xO>0, or, <0.