Abel's theorem in problems and solutions - School of Mathematics

84 Chapter 2
294. Suppose the variation of the argument along a curve be
equal to and be the continuous image of the curve under the
mapping Find the variation of the argument along the curve
295. Find the branch points of the function
In §2.5 we have used the notation
and we have considered some properties of this complex number.
296. Suppose that a curve does not pass through the point
and that is one of the continuous images of the curve under
the mapping Find all the continuous images of the curve
under the mapping
Suppose that two curves and join a point to a different point
Exactly as in the case of function (cf., 284), one proves that if
the curve does not turn around the point then the function
is uniquely defined by continuity along the curves and In this
way if, as for the function we make a cut from the point to
infinity the image of the function turns out to be decomposed
into single-valued continuous branches.
297. Make a cut from the point to infinity, not passing through
the point and define the single-valued continuous branches of the
function by the conditions: where takes the integer
values from 0 to How are the branches expressed in terms of
298. Draw the scheme of the Riemann surface of the function
299. Find the branch points and draw the scheme of the Riemann
surface for the function
300. Find the branch points and draw the scheme of the Riemann
surface of the function
When a multi-valued function has several branch points, in order to
separate the single-valued continuous branches we make the cuts from
every branch point to infinity along lines which do not intersect each
other.
In this way the scheme of the Riemann surface of a given function
may depend essentially on the choice of the cuts from the branch points