Abel's theorem in problems and solutions - School of Mathematics
Abel's theorem in problems and solutions - School of Mathematics
Abel's theorem in problems and solutions - School of Mathematics
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80 Chapter 2<br />
From the solution <strong>of</strong> Problem 279 we obta<strong>in</strong> that also by fix<strong>in</strong>g the<br />
image <strong>of</strong> the <strong>in</strong>itial po<strong>in</strong>t <strong>of</strong> the curve C the cont<strong>in</strong>uous image <strong>of</strong> the<br />
curve C under the mapp<strong>in</strong>g may be def<strong>in</strong>ed non-uniquely. The<br />
uniqueness is lost when the curve C passes through the po<strong>in</strong>t In<br />
fact, for the function the uniqueness is lost only <strong>in</strong> this case,<br />
because only <strong>in</strong> this case do the two images <strong>of</strong> the po<strong>in</strong>t melt <strong>in</strong>to<br />
one po<strong>in</strong>t.<br />
To avoid the non-uniqueness <strong>in</strong> the def<strong>in</strong>ition <strong>of</strong> the images <strong>of</strong> curves<br />
under the mapp<strong>in</strong>g we may exclude the po<strong>in</strong>t <strong>and</strong> forbid<br />
any curve to pass through this po<strong>in</strong>t. This restriction, however, does not<br />
always allows the cont<strong>in</strong>uous s<strong>in</strong>gle-valued branches <strong>of</strong> the function<br />
to be separated.<br />
Indeed, if we fix at a po<strong>in</strong>t one <strong>of</strong> the values <strong>and</strong> we<br />
try to def<strong>in</strong>e at a certa<strong>in</strong> po<strong>in</strong>t by cont<strong>in</strong>uity along two dist<strong>in</strong>ct<br />
curves jo<strong>in</strong><strong>in</strong>g <strong>and</strong> we may obta<strong>in</strong> different values <strong>of</strong> (for<br />
example, see 278). Observe now how we can avoid the non-uniqueness <strong>of</strong><br />
the obta<strong>in</strong>ed value.<br />
280. Suppose the variation <strong>of</strong> the argument <strong>of</strong> along a curve C be<br />
equal to F<strong>in</strong>d the variation <strong>of</strong> the argument <strong>of</strong> along an arbitrary<br />
cont<strong>in</strong>uous image <strong>of</strong> the curve C under the mapp<strong>in</strong>g<br />
281. Let <strong>and</strong> choose Def<strong>in</strong>e the value <strong>of</strong><br />
by cont<strong>in</strong>uity along: a) the segment jo<strong>in</strong><strong>in</strong>g po<strong>in</strong>ts <strong>and</strong><br />
b) the curve with the parametric equation<br />
c) the curve with the parametric equation<br />
282. Let <strong>and</strong> choose at the <strong>in</strong>itial po<strong>in</strong>t <strong>of</strong> a curve C,<br />
Def<strong>in</strong>e by cont<strong>in</strong>uity along the curve C the value <strong>of</strong><br />
at the end po<strong>in</strong>t if the curve C has the equation: a)<br />
b) c)<br />
283. Let C be a closed curve on the plane (i.e., Prove<br />
that the value <strong>of</strong> the function at the end po<strong>in</strong>t <strong>of</strong> the curve C, def<strong>in</strong>ed<br />
by cont<strong>in</strong>uity, co<strong>in</strong>cides with the value at the <strong>in</strong>itial po<strong>in</strong>t if <strong>and</strong> only if<br />
the curve C wraps around the po<strong>in</strong>t an even number <strong>of</strong> times.<br />
In the sequel it will be convenient to use the follow<strong>in</strong>g notation:<br />
DEFINITION. Let C be a cont<strong>in</strong>uous curve with a parametric equation<br />
We shall denote by the curve geometrically identical to C but<br />
oriented <strong>in</strong> the opposite direction; its equation is (cf., 247)