Special delimited oscillators that maxima can't solve it.
Subject: Special delimited oscillators that maxima can't solve it.
From: J.C. Pizarro
Date: Tue, 15 Apr 2008 04:28:50 +0200
On 2008/4/15, J.C. Pizarro <jcpiza at gmail.com>, i wrote:
> 4. They have some symmetries except the functions 5. and 6.
>
> About symmetries, they are odd functions {f(-x) = -f(x)} the 1.,
> even functions {f(-x) = f(x)} the 2., 3., 4., 7. and 8., and they have not
> any symmetry relation the 5. and 6.
>
> 5. The defined integral of these sines and cosines in range [-inf,+inf] is zero
> for odd functions and 2 * defined integral in [0,+inf] for even functions.
sin(1/x) is an odd function and its def.integral in [-inf,+inf] except
in 0 is zero.
cos(1/x) is an even function and its def.integral in [-inf,+inf] except in 0 is
2 * def.integral of cos(1/x) in [0+,+inf].
(%i1) limit(integrate(sin(1/x),x,-y,y),y,%inf);
(%o1) 0
(%i2) integrate(sin(1/x),x,-%inf,%inf);
(%o2) 0
It's OK but with an exception (for all x != 0) (in x=0, its result is
undefined).
(%i3) limit(integrate(cos(1/x),x,-y,y),y,%inf);
y + %inf
/
[ 1
(%o3) limit I cos(-) dx
y -> 0 ] x
/
- y - %inf
(%i4) integrate(cos(1/x),x,-%inf,%inf);
%inf
/
[ 1
(%o4) I cos(-) dx
] x
/
- %inf
Why is not equal it to below?
(%i5) 2*integrate(cos(1/x),x,0,%inf);
%inf
/
[ 1
(%o5) 2 I cos(-) dx
] x
/
0