Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 1) Basic Notions [SI
Putting *=1 and replacing y by ~ ,
SL\
*'
1
we will have
thai is,
/(l. )=f(*. 0).
2. Domain of definition of a function. By the domain of definition of a
function ? f(x, y) we understand a set of points (*, r/) in an jq/-plane in
which the given function is defined (that is to say, in which it takes on definite
real values) In the simplest cases, the domain of definition of a function
U a finite or infinite part of the jo/-plane bounded by one or several
curves (the boundan, of the domain).
Similarly, for a function of three variables u = f(x, y, z) the domain of
definition of the function is a volume in At/z-space.
Example 3. Find the domain of definition of the function
Solution. The function has real values if 4 x 2
y 2 > or x* + y 2 The latter inequality is satisfied hy
< 4.
the coordinates of circle of radius 2 with centre at the coordinate origin.
points lying inside a
The domain of definition
oi the function is the interior of the circle (Fig 64).
Example
1
Fig. 64 Fiy 65
4. Find the domain of definition of the function
z = arc sin + |/ xy
Solution. The first term of the function is defined for 1