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
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
64 Chapter 2<br />
236. Let be a complex number (or, as a particular case, real). Prove<br />
that the complex (or real) function is cont<strong>in</strong>uous for all values<br />
<strong>of</strong> the argument.<br />
237. Prove that the function <strong>of</strong> a complex argument <strong>and</strong><br />
the function <strong>of</strong> a real argument are cont<strong>in</strong>uous for all values <strong>of</strong><br />
their argument.<br />
238. Prove that the function <strong>of</strong> a complex argument is<br />
cont<strong>in</strong>uous for the all values <strong>of</strong><br />
DEFINITION. Let <strong>and</strong> be two functions <strong>of</strong> a complex (or<br />
real) argument. One calls the sum <strong>of</strong> the functions <strong>and</strong> the<br />
function <strong>of</strong> a complex (or real) argument which satisfies at every<br />
po<strong>in</strong>t the equation If the value or the<br />
value is not def<strong>in</strong>ed then the value is also not def<strong>in</strong>ed. In the<br />
same way one def<strong>in</strong>es the difference, the product, <strong>and</strong> the quotient <strong>of</strong> two<br />
functions.<br />
239. Let be a function <strong>of</strong> a complex or real argument <strong>and</strong> let<br />
be cont<strong>in</strong>uous at Prove that at the functions: a)<br />
b) c) are cont<strong>in</strong>uous.<br />
From the result <strong>of</strong> Problem 239(b) we obta<strong>in</strong>, <strong>in</strong> particular, that if<br />
a function is cont<strong>in</strong>uous at a po<strong>in</strong>t <strong>and</strong> is an <strong>in</strong>teger, then the<br />
function is also cont<strong>in</strong>uous at the po<strong>in</strong>t<br />
240. Let <strong>and</strong> be two functions <strong>of</strong> complex or real argument,<br />
cont<strong>in</strong>uous at <strong>and</strong> suppose that Prove that at the<br />
functions: a) b) are cont<strong>in</strong>uous.<br />
DEFINITION. Let <strong>and</strong> be two functions <strong>of</strong> a complex or<br />
real argument. One calls the composition <strong>of</strong> the functions <strong>and</strong><br />
the function which satisfies at every po<strong>in</strong>t the equation<br />
If the value is not def<strong>in</strong>ed, or the function is not<br />
def<strong>in</strong>ed at the po<strong>in</strong>t then the value is also not def<strong>in</strong>ed.<br />
241. Let <strong>and</strong> be two functions <strong>of</strong> complex or real argument.<br />
Let <strong>and</strong> let functions <strong>and</strong> be cont<strong>in</strong>uous at the<br />
po<strong>in</strong>ts <strong>and</strong> respectively. Prove that the function is<br />
cont<strong>in</strong>uous at the po<strong>in</strong>t<br />
From the results <strong>of</strong> Problems 239–241 it follows, <strong>in</strong> particular, that<br />
any expression obta<strong>in</strong>ed from any functions <strong>of</strong> one complex (or real) argument,<br />
cont<strong>in</strong>uous for all values <strong>of</strong> the argument, by means <strong>of</strong> the op-