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.

~ 199. 200.

Investigation of the critical point.

377

- =-( - a) (') + (Z )2 (d) + I3 (- a)3(d ) +
(d\ L ( - =2- 13 a): 3
a,b a,b a,b
is valid for the neighbourhood of the point, as was developed in
~ 188.
The derivates d are obtained from the formula dw - -,
dO dx
~~~~d 0zk,1~~~d z
by successive differentiation. In particular, if fio, f2,o, * * f-1,,0
all vanish at the point z= a, w =- b, the expansion begins with
the term:
w - b- = -L ( ) (-a ) + ~ ' d _ =- fk,o
lk d Zk 7 * d fo,
a,b
If for illustration we consider the relation as one between an
ordinate (w) and an abscissa (z), the tangent to the algebraic curve
at the point =- a, w = b, is then parallel to abscissae and has a contact
of the order k - 1; it meets the curve in k consecutive points.
The point z = a, w = b is a critical point, when for it
fo, =- 0. We proceed to examine what expansions are then valid.
As a first case we have to consider the critical point when fi,O
is not zero. Then S can be expanded by integer powers of w in the
manner just described, and assuming, in order to mention at once
the most general eventuality, that all the partial derived functions
fo,2, fo,3,. * fOk- also vanish for the critical point, we obtain the
expansion:
1 (l1 I k^1 4 /
(w-b11 d b S+-6)kl_ + d
a,b a,b
which we may write:
(z - a) = (w - b) a {1 +, (w - b) + a2 (w - b)2 +... B (w - b)n},
(Lim R =- 0).
Extracting the kth root on both sides, and arranging the right
in powers of (w - b) by means of the Polynomial Theorem, we find:
(z — a) (-,'(w ) - b){l -b)(w —b)2+..T( —b)n } ( LimR'=-)
Denoting (- a) by t, we have now to solve the problem discussed
in ~ 192: to invert the series
t =(w -- b)i 1 + '(w - b) + a'(w b)2 +.
By this means we find an expansion of the form:
w - b = t+ [It + 33t3 +...