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.

Contents.

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Fourth Book. (Pp. 321-389.)
Integrals of complex functions. General properties of analytic functions.
Page
First Chapter: The definite integral of a unique analytic
functi in the complex domain............. 321
~ 177. Definition of the complex integral. 178. Its properties.
179. Condition that the integral be independent of the path of integration.
180. Substitution. 181. The integral within a multiply connected domain.
Condition that it be finite. 182. Examples: Integral 1) of a rational
integer function; 2) of a fractional rational function; 3) of an infinite
series of powers. The singular point.
Second Chapter: Expansion of unique analytic functions in
series of powers. General properties......... 341
~ 183. The problem of calculation. 184. Cauchy's theorem. 185. The
higher derivates. 186. Taylor's series. 187. Examples: The exponential;
the logarithm; the binomial. 188. The implicit algebraic function.
189. Singular points and expansion in their neighbourhood. 190.
Functions classified according to the nature of their singular points.
191. Inversion of the unique analytic function. 192. Lagrange's series.
Third Chapter: Expansion of ambiguous analytic functions,
specially of the algebraic function........... 363
~ 193. The n-valued analytic function. 194. Branching point of the
order (m - i) and integration round it. 195. Expansion in the
neighbourhood of the branching point. 196. Singular points in the
domain of expansion. 197. The branching point also a singular point.
198. The point infinity a branching point. 199. The irreducible algebraic
function defined by the nature of its singular and branching points.
of
200. Investigation of the critical point, when = 0. 201. Simplest case
of the multiple point. 202. Newton's rule for forming the expansion at
an arbitrary point. 203. Example.
Corrections............................... 390
Index............................... 391- 404