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.

~ 48-50.

Circular series.

49. In the series for sinx and tan-'x only odd powers of x
occur, in that for cosx only even powers including zero. The first
two functions are therefore characterised as odd, the other as even.
An odd function f(x) can in general be defined by the property:
that f(x) = - f(- x); for an even function we have f(x) = f(- x).
Thence follows that an odd function, provided it is continuous for
x= 0, must there vanish; it follows further by differentiation that
all its derivates of odd order are even functions, on the other hand
its even derivates are odd functions. Therefore these latter must also
all vanish at the point zero. In the case of even functions on the
other hand, all odd derivates are odd functions and vanish for x = 0.
50. The development of Taylor's series is based on the formation
of the nth derivate. This marks the limit of its applicability; for, if for
any function the general expression of this derivate be too complicated,
the method loses in practicability. Thus, it is not hard to calculate
in general from the recurring formula established for sin-'x the values
of the derived functions for x =0, but that formula is not suited for
forming the remainder.*) Therefore our first endeavour must be to
decide as to the developability of a function and the convergence
of the series of powers from the properties of the function itself
exclusively, not also taking account as heretofore of the properties of
all its derived functions. But then we shall recognise that a series of
powers obtained in any way for a function in an interval must be
identical with the series of Taylor, because f(x) cannot be expressed
by two different series of powers. The investigations require - for
completeness - the extension of the domain of number and for this
new conceptions must first be introduced by the theory of functions
with more than one independent variable.
*) For the expansion of sin-ix in a series, see Integral Calculus, Bk. III
Chap. IV.