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.

~ 111-112. Partial fractions in case of simple conjugate roots.

189

The logarithm of a complex quantity is many-valued, but as its
values differ only by an additive imaginary constant, it is indifferent
in the indefinite integral which of its values is attributed to each
individual logarithm. The transition to the definite integral by formation of the difference of two values of the function, requires that
the value of the function at the one limit shall proceed continuously
from its value at the other limit; compare the end of next Section.
112. If the coefficients of (p(x) be real, it may have complex
roots but they are conjugate in pairs (~ 90). When it has, we can
prevent complex values appearing in the final formula, provided the
coefficients of g are also real, by combining the partial fractions
relative to conjugate complex roots.
Let a -+ i3 and a - if be two conjugate roots, then
-( + it) is a complex quantity: + -iN,
p'(, -+ if)
and ((O — it) is the conjugate value: M- iN;
cpq(c - ip)
therefore:
M +- i X M-N 2M(x -a) - 2Nv
x - (a + ip) + x - ( - ip) (X - a)2 + P2
is a real quotient. The constants of the numerator can also be directly
determined, by starting from the identity:
^Mx) -- Px +Q ~1P(X)2
9(x) (- (x )2P + t 1(x) where 9(x) (x - c2 + 2) i(x),
hence:
(x) = (Px + Q) 1, (x) + (x- 2~ + P) 1p (x).
Substituting for x the two values a + ift we have the equations:
P(ac i ip) + Q (e + ip)
whose real and imaginary parts give two equations to determine
P and Q.
The integral:
Px+Q dx
J (x - - p+ 2 -can be resolved into the integrals:
p x p- a) dx dx
X J - ')2 + P-Pa Q) ( x — j U) + P2
Putting x- ~2 + 2' -/, (x - c) dx x - d z, the first of these becomes:
rpf (x - )clx P dz 2 p ) (x - 2 + p2)
(xI U?4, —s z =22 25 - cc
The second can be reduced, by putting x - a = 3pz, dx = fldz, to a
Fundamental Integral: