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.

~ 189. Singular points and expansions in their neighbourhoods.

353

fax) 4_ f x- ff
JYf- dz) =J dz — J- - dc -— ff - d... -j dz.
(t) (a) (c,) (cz) (c,)
Now on the left side of this equation we have the integral:
f (z-)d = 2izf(t).
(
But the integrals on its right can be expressed by series. Since for
the first of them S signifies a point upon the circle R, we have
mod [ - a] > mod [t - c]; therefore:
z - t z - a_ t -o c - t. - z (t2 -(Z o)3
Z- a.
is an absolutely convergent series. Accordingly:
VIfJ -rf- f- a d { + (d(t -- (- - 3 dz + *
(a) (a) (a) (a)
but the coefficients of this series are no longer the derived functions
at the point a as before, although they have the same form; for
the circle of radius R, round which these integrals are formed, can no
longer be arbitrarily contracted about the point a.
For each integral round a point c: mod [t - c] > mod [- c]
because t is a point outside its circle O; consequently:
1 _ -1 1 __ 1 f C,2~C (- C2 )
t —b t-c' zt-c ltci + -c (t-c)2+ (zt-c)3 +a
}
t —~
is an absolutely convergent series; and we have: - f () ds —
~II~)CE1~+1. q-x~j 1 + fz)dz-tVII. + f( +t + -c)SJ ' - )((t C)j ( ( -)2( + etc..
+ c) (C) (C)
Denoting briefly the coefficients of series VI. divided by 2iz by
A0, A, A2,..;; likewise those of series VII., referring to the point
Ck, by Cl(k), C2(k), C3(k),.., and in this the definite integrals can be
taken along a circle of arbitrarily small radius; the result is: For all
points t within the circle R, in which c, c2, 2... Cn are singular
points for the unique analytic function f(z), we have the expansion:
VIII. f(t) Ao +- A, (t - a) + A, (t - )2 + A3(t — a)3 +..
C C_ C' __
t-c c (t- c)2 (t- )3 (t - c1)
+( l (n) C (n) C, (n)
+ -- + C2 +) + ~ — + ''
t -C (t - C)2 (t - cn)3 (t - c4)4
in series proceeding both by positive and negative integer powers of t.
HARNACK, Calculus. 23