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Unformatted text preview: e that x and y are real numbers that are not both zero and that
arccos x
x2 y2 . We see that
sin 1 cos 2 1 cos 2 arccos x2 x
y2 2
We define r x
x2 y2 1 x 2 y 2 and we observe that 380 y
x2 y2 . x r cos
and
y r sin .
In the event that the latter equation says that y r sin we define and if the equation
y r sin is false (in which case y r sin ) we define . In either event we have the two
equations
x r cos
y r sin
Finally we observe that since 0 we must have 0 2. However, if then the
equation y r sin is true and . Thus the case 2 cannot occur and we conclude that
0 2. Exercises on Analytic Functions
Prove that the function arctan is analytic on R.
Suppose that c is any number. Using the fact that the rational function whose value at every
number x is 1 2 is analytic, choose 0 and a sequence a n such that
1 x 1
1 x2
whenever x Ý an x
Ý arctan c
n0 for x n n0 c  . If we now define
f x arctan x c an x
n1 c n 1 c  1 then we see at once that f is the constant 0 and so
Ý arctan x arctan c
n0 whenever x an x
n1 c n 1 c  . 1. Given that
f x 3x
for every number x, prove that f is analytic on R
0 but is not analytic on R.
Suppose that c 0. The equation
f x cx c 1/3 c 1/3 1 x c c 1/3 Ý c 1/3
n0 1
cn n x c n holds whenever x c  c . Therefore f is analytic on the set R
0 . Since f is not differentiable at
0, the function f can’t be analytic on any open interval that contains 0.
2. Prove that the function tan is analytic in some neighborhood of the number 0.
The function tan, being the quotient of the two analytic functions sin and cos, is analytic on the
interval
,
.
22
3. Given that
fx exp
0 1
x2 if x 0
, if x 0 prove that f is analytic on R
0 but is not analytic on R.
In spite of the fact that this function f has derivatives of all orders at every number, we saw in an 381 earlier example that f can’t be expressed as the sum of it’s Taylor series center 0 in any
neighborhood of 0. On the other hand, the composition theorem for analytic functions guarantees
that f is analytic on the set R
0.
4. Prove that a rational function is analytic on any open set in which its denominator does not vanish.
This fact follows at once from the fact that the quotient of two analytic functions is analytic as long
as the denominator is not zero.
5. Given that
1x fx 1/x if x 0 if x 0 e prove that f is analytic on the interval 1, 1 . Use Scientific Notebook to work out the first few terms of some
of the series expansions of this function. To do so, point at the expression 1 x 1/x , open the Maple menu
and click on the Power Series option.
We begin by observing that
Ý 1 n n 1
x
n1 log 1 x
n0 whenever x  1. We now define Ý 1n n
x
n1 gx
n0 and we observe that, since g 0 1 and
gx
for every x 1, 1 log 1 x
x 0 we have
f x exp g x whenever x  1. Since g is analytic on the interval
funct...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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